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≥ 0
(3.2)
for all x ∈ D(Ak ). Definitizable operators have a spectral decomposition, which is similar to the spectral theorem in Hilbert spaces, except that there is in general an additional finite point spectrum in the complex plane (see [3, p. 180], [16, Thm 3.2, p. 34] and Lemma 3.3 below). An important special case of a Krein space is when K is positive except on a finite-dimensional subspace, i.e. κ := dim K− < ∞.
(3.3)
In this case the Krein space is called a Pontrjagin space of index κ. Classical results of Pontrjagin (see [3, Thms 7.2 and 7.3, p. 200] and [16, p. 11–12]) yield that any selfadjoint operator A on a Pontrjagin space is definitizable, and that it has a κ-dimensional negative subspace which is A-invariant. We now explain how the abstract theory applies to the wave equation in the Kerr geometry. In order to have a spectral theorem, the Hamiltonian must be definitizable. There is no reason why H should be definitizable on the whole space (r1 , ∞) × S 2 , and this leads us to consider the wave equation in “finite volume” [rL , rR ] × S 2 with Dirichlet boundary conditions. Thus setting = ( , i t ) and regarding the two components ( 1 , 2 ) of as independent functions, we consider the vector space PrL ,rR = (H 1,2 ⊕ L2 )([rL , rR ] × S 2 ) with Dirichlet boundary conditions 1 (rL ) = 0 = 1 (rR ) .
(3.4)
Our definition of H 1,2 ([rL , rR ]×S 2 ) coincides with that of the space W 1,2 ((rL , rR )×S 2 ) in [11, Sect. 7.5]. Note that we only impose boundary conditions on the first component 1 of , which lies in H 1,2 . According to the trace theorem [7, Part II, Sect. 5.5, Theorem 1], the boundary values of a function in H 1,2 ([rL , rR ] × S 2 ) are in L2 (S 2 ), and therefore we can impose Dirichlet boundary conditions. We endow this vector space with the inner product associated to the energy; i.e. in analogy to (2.13), < , > =
rR
dr rL
1 −1
d(cos ϑ)
(r 2 + a 2 )2 − a 2 sin2 ϑ 2 2
+ ∂r 1 ∂r 1 + sin2 ϑ ∂cos ϑ 1 ∂cos ϑ 1
1 a2 + ∂ϕ 1 ∂ ϕ 1 . − sin2 ϑ
(3.5)
Lemma 3.1. For every rR > r1 there is a countable set E ⊂ (r1 , rR ) such that for all rL ∈ (r1 , rR ) \ E, the inner product space PrL ,rR is a Pontrjagin space. The topology on PrL ,rR is the same as that on (H 1, 2 ⊕ L2 )([rL , rR ] × S 2 ).
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Proof. Since (3.5) involves no terms which mix the first component of with the second component, PrL ,rR clearly has an orthogonal direct sum decomposition PrL ,rR = V1 ⊕V2 with V1/2 = { ∈ PrL ,rR : 2/1 ≡ 0}. Furthermore, it is obvious that the space (V2 , <., .> ) has a positive scalar product and that the corresponding norm is equivalent to the L2 -norm. Hence it remains to consider V1 , i.e. the space H 1,2 ([rL , rR ] × S 2 ) with Dirichlet boundary conditions and the inner product < , > =
1 dr d(cos ϑ) rL −1 × ∂r ∂r + sin2 ϑ ∂cos ϑ ∂cos ϑ
a2 1 2 − . + k sin2 ϑ rR
(3.6)
Transforming to the variable u, (2.18), and using the representation (2.23), one sees that on the subspace C 2 ([uL , uR ] × S 2 ) the inner product (3.6) can be written as < , > = ( , A )L2 ([uL ,uR ]×S 2 ,dµ)
(3.7)
with A according to (2.22). Here we set uL = u(rL ), uR = u(rR ), and dµ is the measure (2.26). A is a Schr¨odinger operator with smooth potential on a compact domain. Standard elliptic results [20, Proposition 2.1 and the remark before Proposition 2.7] yield that A is essentially selfadjoint in the Hilbert space H = L2 ([uL , uR ] × S 2 , dµ). It has a purely discrete spectrum which is bounded from below and has no limit points. The corresponding eigenspaces are finite-dimensional, and the eigenfunctions are smooth. Let us analyze the kernel of A. Separating and using that the Laplacian on S 2 has eigenvalues −l(l + 1), l ∈ N0 , A has a non-trivial kernel if and only if for some l ∈ N0 , the solution of the ODE ∂ ∂ a2 k2 φ(u) = 0 (3.8) − (r 2 + a 2 ) + 2 l(l + 1) − ∂u ∂u r + a 2 r 2 + a2 with boundary conditions φ(uR ) = 0 and φ (uR ) = 1 vanishes at u = uL . Since this φ has at most a countable number of zeros on (−∞, uR ] (note that φ(u) = 0 implies φ (u) = 0 because otherwise φ would be trivial), φ vanishes at uL only if uL ∈ El with El countable. We conclude that there is a countable set E = ∪l El such that the kernel of A is trivial unless uL ∈ E. Assume that uL ∈ / E. Then A has no kernel, and so we can decompose H into the positive and negative spectral subspaces, H = H+ ⊕ H− . Clearly, H− is finite-dimensional. Since its vectors are smooth functions, we can consider H− as a subspace of PrL ,rR , and according to (3.7) it is a negative subspace. Its orthogonal complement in PrL ,rR is contained in H+ and is therefore positive. We conclude that PrL ,rR is positive except on a finite-dimensional subspace. It remains to show that the topology induced by <. , .> is equivalent to the H 1,2 topology. Since on finite-dimensional spaces all norms are equivalent, it suffices to consider for any λ0 > 0 the spectral subspace for λ ≥ λ0 , denoted by Hλ0 . We choose λ0 such that a2 k2 1 − λ0 ≤ V0 := min − 2 <0. [rL ,rR ] r + a2
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F. Finster, N. Kamran, J. Smoller, S.-T. Yau
Then for every ∈ C 2 ∩ Hλ0 , (∗)
< , > = , A L2 (dµ) ≤ c 2H 1,2 , λ0 1 < , > ≥ , A L2 (dµ) + 2L2 (dµ) 2 2 (∗) 1 V0 − 1 λ0 1 2 ≥ H 1,2 + 2L2 (dµ) + 2L2 (dµ) ≥ 2H 1,2 , 2c 2 2 2c where in (∗) we used that the coefficients of the ODE (3.8) are bounded from above and below and that the zero order term is bounded from below by V0 . We always choose rL and rR such that PrL ,rR is a Pontrjagin space and that our initial data is supported in [rL , rR ] × S 2 . We now consider the Hamiltonian (1.13) on the Pontrjagin space PrL ,rR with domain C ∞ ([rL , rR ] × S 2 )2 ⊂ PrL ,rR . For clarity, we often denote this operator by HrL ,rR . Lemma 3.2. HrL ,rR has a selfadjoint extension PrL ,rR . Proof. On the domain of HrL ,rR , the scalar product can be written in analogy to (2.23) as < , > = ( , S )L2 ([uL ,uR ]×S 2 ,dµ) , where the operator S acts on the two components of as the matrix A0 S= , 0 1
(3.9)
where A is again given by (2.22) and dµ is the measure (2.26). As shown in Lemma 3.1, S has a selfadjoint extension and is invertible. We introduce on C0∞ ([uL , uR ] × S 2 )2 the 1
1
operator B by B = |S|− 2 SH |S|− 2 . The fact that H is symmetric in PrL ,rR implies that B is symmetric in L2 ([uL , uR ] × S 2 , dµ). A short calculation shows that 1 |A| |A|− 2 Aβ 2 . B = 1 β|A|− 2 A |A| + β 2
Treating the terms involving β as a relatively compact perturbation, we readily find that B 2 is selfadjoint on L2 ([uL , uR ]×S 2 , dµ) with domain D(B 2 ) = D(A). Consequently, 1 the spectral calculus gives us a selfadjoint extension of B with domain D(B) = D(A− 2 ). 1 We extend H to the domain D(H ) := |S|− 2 D(B). We now show that with this new domain, H is selfadjoint on PrL ,rR . Suppose that for some vectors , ∈ PrL ,rR , < , H >=< , >
for all ∈ D(H ).
It then follows that 1
1
1
1
(|S| 2 , B|S| 2 )L2 (dµ) = (S|S|− 2 , |S| 2 )L2 (dµ)
for all ∈ D(H ) 1
1
(note that the vectors , ∈ PrL ,rR lie in H 1,2 ⊕ L2 and thus both |S| 2 , S|S|− 2 ∈ L2 (dµ)). By definition of the domain of H , this is equivalent to ˜ L2 (dµ) = (S|S|− 2 , ) ˜ L2 (dµ) (|S| 2 , B ) 1
1
˜ ∈ D(B). for all
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Since B is selfadjoint, it follows that the vector |S| 2 lies in the domain of B and that 1 1 B|S| 2 = S|S|− 2 . This implies that ∈ D(H ) and that H = . We now prove a basic lemma on the structure of the spectrum of the Hamiltonian HrL ,rR . Lemma 3.3. The spectrum of HrL ,rR is purely discrete. It consists of finitely many complex spectral points appearing as complex conjugate pairs, and of an infinite sequence of real eigenvalues with no accumulation points. Proof. Since the operator HrL ,rR is essentially selfadjoint, there exists a negative definite subspace L− of PrL ,rR of dimension κ which is HrL ,rR -invariant (see [16, p. 11]). Let p0 denote the minimal polynomial of HrL ,rR on L− , i.e. p0 (HrL ,rR ) L− = 0 with deg p0 ≤ κ minimal. Furthermore, we let p be the real polynomial of degree ≤ 2κ defined by p = p0 p0 . We claim that im p (HrL ,rR ) is a positive semi-definite subspace. Indeed, we have for all x ∈ PrL ,rR ,
(3.10)
im p (HrL ,rR ) ⊂ im p0 (HrL ,rR ) ⊂ (L− )⊥ ⊂ (PrL ,rR )+ ,
(3.11)
so that
as claimed. Next, since the square of the operator HrL ,rR , is elliptic it follows that dim ker (p (HrL ,rR ) < ∞,
(3.12)
and from [16, Prp. 2.1], we know that for each eigenvalue ξ of HrL ,rR , the corresponding Jordan chain has finite length bounded by 2κ + 1. It follows that p 2κ+1 (HrL ,rR ) has a finite-dimensional kernel and no Jordan chains. This implies that im p2κ+1 (HrL ,rR ) ∩ ker p 2κ+1 (HrL ,rR ) = {0} .
(3.13)
Furthermore, since the operator p 2k+1 (HrL ,rR ) is selfadjoint, its image and kernel are clearly orthogonal. The image of p2k+1 (HrL ,rR ) is contained in im p (HrL ,rR ) and is therefore positive semi-definite. We shall now show that the space im p 2κ+1 (HrL ,rR ) is actually positive definite. To this end, we let N be its null space, N := {x ∈ im p2κ+1 (HrL ,rR ), <x , x>= 0}.
(3.14)
For all x ∈ N and y ∈ D(HrL ,rR ), we have <x , p2κ+1 (HrL ,rR ) y>= 0,
(3.15)
(3.16)
which is equivalent to
because p is real. Since the scalar product is non-degenerate, this implies that p 2κ+1 (HrL ,rR ) x = 0 .
(3.17)
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F. Finster, N. Kamran, J. Smoller, S.-T. Yau
But we have just shown that ker p 2κ+1 (HrL ,rR ) and im p 2κ+1 (HrL ,rR ) have trivial intersection. It follows that x = 0 and therefore that im p 2κ+1 (HrL ,rR ) is positive definite, as claimed. Restricting HrL ,rR to im p 2κ+1 (HrL ,rR ), we have a self-adjoint operator on a Hilbert space. Thus the spectral theorem in Hilbert space applies, and the ellipticity of Hr2L ,rR yields that the spectrum is purely discrete. On the finite-dimensional orthogonal complement ker (p 2κ+1 (Hua ,ub )) we bring HrL ,rR into the Jordan canonical form. 4. Resolvent Estimates In this section we consider the Hamiltonian H as a non-selfadjoint operator on the Hilbert space H with the scalar product (., .) according to (2.16). We work either in infinite volume with domain of definition D(H ) = C0∞ ((r1 , ∞) × S 2 )2 or in the finite box r ∈ [rL , rR ] with domain of definition given by the functions in C ∞ ((rL , rR ) × S 2 )2 which satisfy the boundary conditions (3.4). Some estimates will hold in the same way in finite and infinite volume. Whenever this is not the case, we distinguish between finite and infinite volume with the subscripts rL ,rR and ∞ , respectively. We always consider a fixed k-mode. The next lemma shows that the operator H − ω is invertible if either |Im ω| is large or |Im ω| = 0 and |Re ω| is large. The second case is more subtle, and we prove it using a spectral decomposition of the elliptic operator A which generates the energy scalar product. This lemma will be very useful in Sect. 7, because it will make it possible to move the contour integrals so close to the real axis that the angular estimates of Lemma 2.1 apply. By a slight abuse of notation we use the same notation for H and its closed extension. Lemma 4.1. There are constants c, K > 0 such that for all ∈ D(H ) and ω ∈ C, 1 K (H − ω) ≥ |Im ω| − . c 1 + |Re ω| Proof. For every unit vector ∈ D(H ), (H − ω) ≥ |( , (H − ω) )| ≥ |Im ( , (H − ω) )| 1 ≥ |Im ω| − ( , (H − H ∗ ) ) . 2
(4.1)
It is useful to work again in the variable u and the representation (2.24) of the scalar product (., .) on C02 (R × S 2 )2 . We introduce on C02 (R × S 2 )2 the operator H+ by H+ =
0 1 A+δ β
.
Comparing with (2.24) one sees that H+ is formally selfadjoint w.r. to the scalar product (., .). Furthermore, one sees from (2.25) that H+ differs from H only by a bounded operator, 0 0 H − H+ = −δ 0 ≤ c .
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Thus on C 2 (R × S 2 )2 , H − H ∗ = (H − H+ ) − (H − H+ )∗ ≤ H − H+ + (H − H+ )∗ = 2 H − H+ ≤ 2c , and substituting this bound into (4.1), we conclude that (H − ω) ≥ (|Im ω| − c) . In view of this inequality it remains to consider the case where |Re ω| is large. Using standard elliptic theory (see again [20, p. 86, Proposition 2.7] and [5]), the operator A with domain D(A) = C0∞ (R × S 2 ) is essentially self-adjoint on the Hilbert space L2 (dµ) := L2 (R × S 2 , dµ), with dµ according to (2.26). Clearly, A is bounded from below, A ≥ −c, and thus σ (A) ⊂ [−c, ∞). For given 1 we let P0 and P be the spectral projectors corresponding to the sets [−c, 2 ) and [2 , ∞), respectively. We decompose a vector ∈ H in the form = 0 + with P0 0 P 0 , = . 0 = 0 P0 0 P This decomposition is orthogonal w.r. to the energy scalar product, A0 0 L2 (dµ) = 0 . < , 0> = , 0 1 However, our decomposition is not orthogonal w.r. to the scalar product (., .), because A+δ 0 δ0 0 L2 (dµ) = , 0 L2 (dµ) . ( , 0 ) = , 0 1 00 But at least we obtain the following inequality, 1 L2 (dµ) , |( , 0 )| ≤ c 0
(4.2)
1 denotes the first component of . Using that where 1 2 1 1 L2 (dµ) = , A−1 L2 (dµ) ≤
1 2 , 2
we can also write (4.2) in the more convenient form |( , 0 )| ≤
c 0 .
Choosing sufficiently large, we obtain 2 = 2 + 2 Re ( , 0 ) + 0 2 ≤ 4 ( + 0 )2 and thus ≤ 2 ( + 0 ) .
(4.3)
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F. Finster, N. Kamran, J. Smoller, S.-T. Yau
Furthermore, we can arrange by choosing sufficiently large that A0 L2 (dµ) < , > = , 0 1 1 A+δ 0 ≥ , L2 (dµ) 0 1 2 1 = 2 . 2 Next we estimate the inner products < , H 0>, ( 0 , H ) and ( 0 , H 0 ). The calculations 0 A 00 0 L2 (dµ) = , 0 L2 (dµ) , < , H 0> = , Aβ 0β 0 A+δ 0δ ( 0 , H ) = 0 , L2 (dµ) = 0 , L2 (dµ) , A β 0β 0 A+δ |( 0 , H 0 )| = 0 , 0 L2 (dµ) A β
A 01 2L2 (dµ)
≤ c 0 L2 (dµ) + 2 A 01 L2 (dµ) 02 L2 (dµ) , 2 A 0 0 L2 (dµ) = 0 , 0 0 A0 2 ≤ 0 , 0 L2 (dµ) = 2 0 2 0 1
give us the bounds | < , H 0> | ≤ c 0 , |( 0 , H )| ≤ c 0 , |( 0 , H 0 )| ≤ (c + 2) 0 2 . Using the above inequalities, we can estimate the inner product < , (H − ω) > by | < , (H − ω) > | ≥ | < , (H − ω) > | − | < , (H − ω) 0> | |Im ω| ≥ 2 − c 0 . 2 Applying the Cauchy-Schwarz inequality | < , (H −ω) > | ≤ c1 (H −ω) and dividing by , we obtain (possibly after increasing c) that (H − ω) ≥
|Im ω| − 0 . c
Next we estimate the inner product ( 0 , (H − ω) ), |( 0 , (H − ω) )| ≥ |( 0 , (H − ω) 0 )| − |( 0 , (H − ω) )| |ω| ≥ (|ω| − c − 2) 0 2 − c 1 + 0 .
(4.4)
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275
We apply the Cauchy-Schwarz inequality ( 0 , (H − ω) ) ≤ 0 (H − ω) and divide by 0 , |ω| . (H − ω) ≥ (|ω| − c − 2) 0 − c 1 +
(4.5)
Choosing = (|ω| − c)/4 and increasing c, the inequalities (4.4) and (4.5) give for sufficiently large |ω| the bounds |Im ω| − 0 , c |ω| (H − ω) ≥ 0 − c . 2
(H − ω) ≥
Multiplying the second inequality by 4/|ω| and adding the first inequality, we conclude that 4c |Im ω| − + 0 . 2 (H − ω) ≥ c |ω| The result now follows from (4.3).
With the last lemma at hand, we are ready to introduce the resolvent. Namely, we let =
2K ω ∈ C : |Im ω| ≥ 1 + |Re ω|
(4.6)
with K as in Lemma 4.1. Corollary 4.2. If ω ∈ , the operator H − ω is invertible. The corresponding resolvent S(ω) := (H − ω)−1 satisfies the bound S(ω) ≤
c |Im ω|
(4.7)
with c independent of ω ∈ . Proof. In view of the preceding lemma, it suffices to show that the image of H − ω is ˆ ∈ H such that dense in H for any ω ∈ . Otherwise, there would exist a non-zero ˆ = 0 <(H − ω) , >
for all ∈ D(H ) ,
(4.8)
that is a weak solution of the equation (H − ω) = 0. By the regularity theorem for elliptic operators on manifolds with boundary (cf. [19, Chapter 5, Theorem 1.3]), every weak solution of this equation is a solution in the strong sense. These have been ruled out by the preceding lemma.
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Using (4.6) in (4.7), we immediately get the bound S(ω) ≤ c (1 + |Re ω|) .
(4.9)
Since S(ω) is a bounded operator, its domain of definition can clearly be chosen to be the whole Hilbert space. We shall assume until the end of this section that ω ∈ . The next lemma gives detailed estimates for the difference of the resolvents SrL ,rR and S∞ in finite and infinite volume, respectively. By Qλ (ω) we denote a given projector onto an invariant subspace of the angular operator Aω corresponding to the spectral parameter λ of dimension at most N (see Lemma 2.1 for details). Lemma 4.3. For every ∈ C0∞ ((rL , rR ) × S 2 )2 and every p ∈ N, there is a constant C = C( , p) (independent of ω) such that < , Sr
L ,rR
(ω) − S∞ (ω) > ≤
1 C . 1 + |ω|p |Im ω|
(4.10)
Furthermore, for every ∈ C0∞ ((rL , rR ) × S 2 )2 and every p ∈ N and q ≥ N , there is a constant C = C( , p, q) (independent of ω and λ) such that < , Qλ Sr ,r (ω) − S∞ (ω) > ≤ L R
C 1 Qλ . (4.11) (1 + |ω|p )(1 + |λ|q ) |Im ω|
Proof. By definition of the resolvent, (H − ω) S(ω) = . This relation holds both in finite and in infinite volume, and thus (H − ω) SrL ,rR (ω) − S∞ (ω) (r, ϑ) = 0 if rL ≤ r ≤ rR . Iterating this identity and using the fact that H and S commute, we see that on [rL , rR ]×S 2 , ωp+1 SrL ,rR (ω) − S∞ (ω) = SrL ,rR (ω) − S∞ (ω) H p+1 . (4.12) Combining this identity with the Schwarz-type inequality (2.17), we obtain < , Sr ,r (ω) − S∞ (ω) > ≤ c1 Sr ,r (ω) − S∞ (ω) 2 , L R L R p+1 |ω | < , SrL ,rR (ω) − S∞ (ω) > ≤ < , SrL ,rR (ω) − S∞ (ω) H p+1 > , ≤ c1 SrL ,rR (ω) − S∞ (ω) H p+1 . Since is smooth and has compact support, H p+1 also has these properties. The estimate (4.9) gives (4.10). In order to prove (4.11), we first combine (4.12) with (2.17) to obtain (1 + |ω|p+1 ) < , Qλ (SrL ,rR − S∞ ) > ≤ c1 Qλ SrL ,rR − S∞ + H p+1 . (4.13) Since q is at least as large as the dimension of the invariant subspace corresponding to λ, (Aω − λ)q Qλ = 0. Therefore, for every ∈ C0∞ ([rL , rR ] × S 2 )2 , 0 = < , (Aω − λ)q Qλ > = <(A∗ω − λ)q , Qλ > . Expanding the power (A∗ω − λ)q and using (2.17), we obtain
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277
q |λ|q < , Qλ > ≤ cl |λ|q−l (A∗ω )l χ[rL ,rR ] Qλ l=1
with combinatorial factors cl (here χ[rL ,rR ] is the operator of multiplication by the characteristic function). Since the angular operator A∗ω is according to (1.9) a polynomial in ω of degree two, the function (A∗ω )l is also polynomial in ω, i.e. (A∗ω )l
=
2l
ω p p ,
p=0
where the functions p are composed of and its angular derivatives, as well as the coefficient functions of A∗ω . This gives the estimate (A∗ω )l ≤
2l
|ω|p p ≤ c (1 + |ω|2l )
p=0
with a constant c which depends only on and l. We thus obtain q |λ| < , Qλ > ≤ cl ( ) |λ|q−l (1 + |ω|2l ) χ[rL ,rR ] Qλ . q
l=1
Young’s inequality allows us to compensate the lower powers of λ, |λ|q < , Qλ > ≤ c(q, ) (1 + |ω|2q ) χ[rL ,rR ] Qλ . We now choose equal to the left side of (4.12) with p = 0 and p = r and take the sum of the resulting inequalities. Applying again the Schwarz inequality, we obtain |λ|q (1 + |ω|r ) < , Qλ (SrL ,rR − S∞ ) > ≤ c (1 + |ω|2q ) Qλ SrL ,rR − S∞ + H r . By choosing r sufficiently large, we can compensate the factor (1 + |ω|2q ) on the right. More precisely, |λ|q (1 + |ω|p+1 ) < , Qλ (SrL ,rR − S∞ ) > ≤ c Qλ SrL ,rR − S∞ + H p+2q+1 . Adding this inequality to (4.13) and substituting the estimate (4.9) gives (4.11).
5. Separation of the Resolvent In this section we fix ω ∈ σ (H ), so that the resolvent S = (H − ω)−1 exists. As in the previous section, we assume that Qλ is a given projector onto a finite-dimensional invariant subspace of the angular operator Aω corresponding to the spectral parameter λ. Our goal is to represent the operator product Qλ S in terms of the solutions of the radial ODE. According to (1.10) and (1.8), the radial ODE is 2 ∂ ∂ (r 2 + a 2 )2 ak − − ω+ 2 + λ R(r) = 0 , (5.1) ∂r ∂r r + a2 where λ is the separation constant. We can assume that k ≥ 0 because otherwise we reverse the sign of ω. We again work in the “tortoise variable” u, (2.18), and set
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F. Finster, N. Kamran, J. Smoller, S.-T. Yau
φ(r) =
r 2 + a 2 R(r) .
(5.2)
Then Eq. (5.1) can be written as 2 1 φ ∂ λ ak 2 2 ∂ − 2 = 0 . (5.3) (r + a ) + ω+ 2 √ 2 2 2 2 2 r + a ∂u ∂u r +a (r + a ) r 2 + a2 Using that (r 2 + a 2 )
1 1 ∂ 1 ∂ 1 ∂ (r 2 + a 2 )− 2 = − (r 2 + a 2 )− 2 (r 2 + a 2 ) = − (r 2 + a 2 ) 2 , ∂u 2 ∂u ∂u
(5.3) simplifies to the Schr¨odinger-type equation ∂2 − 2 + V (u) φ(u) = 0 ∂u
(5.4)
with the potential
ak V (u) = − ω + 2 r + a2
2 +
λ 1 2 + ∂ r 2 + a 2 . (5.5) √ u (r 2 + a 2 )2 r 2 + a2
We let φ1 and φ2 be two solutions of (5.4) which are compatible with the boundary conditions. More precisely, in finite volume we satisfy the Dirichlet boundary conditions φ1 (uL ) = 0 and φ2 (uR ) = 0 (again with uL = u(rL ) and uR = u(rR )). Likewise, in infinite volume we only consider the case Im ω < 0 and let φ1 and φ2 be the fundamental solutions which decay exponentially at u = −∞ and u = +∞, respectively (the existence of these fundamental solution will be established in Corollary 6.4). If the solutions φ1 and φ2 were linearly dependent, they would give rise to a vector in the kernel of H − ω, in contradiction to our assumption ω ∈ σ (H ). Thus the Wronskian w(φ1 , φ2 ) := φ1 (u) φ2 (u) − φ1 (u) φ2 (u)
(5.6)
is non-zero (note that w is by definition independent of u). We begin by constructing the “Green’s function” corresponding to (5.4). Lemma 5.1. The function
1 φ1 (u) φ2 (u ) if u ≤ u s(u, u ) := × φ2 (u) φ1 (u ) if u > u w(φ1 , φ2 )
(5.7)
satisfies the distributional equation ∂2 − 2 + V (u) s(u, u ) = δ(u − u ) . ∂u Proof. By definition of the distributional derivative, ∞ ∞ η(u) (−∂u2 + V )s(u, u ) du = (−∂u2 + V )η(u) s(u, u ) du −∞
−∞
for every test function η ∈ C0∞ (R). It is obvious from its definition that the function s(., u ) is smooth except at the point u = u , where its first derivative has a discontinuity. Thus after splitting up the integral, we can integrate by parts twice to obtain
Scalar Wave Equation in Kerr Geometry
∞ −∞
=
279
(−∂u2 + V )η(u) s(u, u ) du
u
η(u) (−∂u2 + V )s(u, u ) du + lim η(u) ∂u s(u, u )
−∞ ∞
+
u
uu
η(u) (−∂u2 + V )s(u, u ) du − lim η(u) ∂u s(u, u ) . uu
Since for u = u , s is a solution of (5.4), the obtained integrals vanish. Computing the limits with (5.7), we get ∞ 2 (−∂u + V )η(u) s(u, u ) du = lim − lim η(u) ∂u s(u, u ) −∞
=
uu
uu
1 η(u ) φ1 (u ) φ2 (u ) − φ2 (u ) φ1 (u ) = η(u ) , w(φ1 , φ2 )
where in the last step we used the definition of the Wronskian (5.6).
In what follows we also regard s(u, u ) as the integral kernel of a corresponding operator s, i.e. (sφ)(u) := du s(u, u ) φ(u ) du . If Qλ projects onto an eigenspace of Aω , we see from (1.10), (1.7), and (5.2) that 1 1 (r 2 + a 2 )− 2 Qλ (ϑ, ϑ ) s(u, u ) = (r 2 + a 2 )− 2 Qλ (ϑ, ϑ ) δ(u − u ) . (5.8) Loosely speaking, this relation means that the operator product Qλ s is an angular mode of the Green’s function of the wave equation. Unfortunately, Qλ might project onto an invariant subspace of Aω which is not an eigenspace. In this case, the angular operator has on the invariant subspace the “Jordan decomposition” Aω Qλ = (λ + N ) Qλ
(5.9)
with N = N (ω, λ) a nilpotent operator. Lemma 5.3 extends (5.8) to this more general case. In preparation, we need to consider powers of the operator s. Lemma 5.2. For every l ∈ N0 , the operator s l is well-defined. Its kernel (s l )(u, u ) has regularity C 2l−2 . Proof. Writing out the operator products with the integral kernel, one sees that the operator s l is obtained from s by iterated convolutions, p+1 s (u, u ) = s(u, u ) s p (u , u ) du . (5.10) In the finite box, these convolution integrals are all finite because s(u, u ) is continuous and the integration range is compact. In infinite volume, the function s(u, u ) decays exponentially as u, u → ±∞ (see Corollary 6.4), and so the integrals in (5.10) are again finite. Hence s l is well-defined. Let us analyze the regularity of the integral kernel of s l . By definition, s(u, u ) is continuous, and (5.10) immediately shows that the same is true for s p (u, u ). Differentiating through (5.10) and applying Lemma 5.1, one sees that s p satisfies for p > 1 the distributional equation
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∂2 − 2 + V (u) (s p )(u, u ) = (s p−1 )(u − u ) . ∂u
This shows that incrementing p indeed increases the order of differentiability by two.
Lemma 5.3. For given λ ∈ σ (Aω ) we let g be the operator g =
∞
(−N )l s l+1 ,
(5.11)
l=0
where N is the nilpotent matrix in the Jordan decomposition (5.9). Then 1 1 (r 2 + a 2 )− 2 Qλ (ϑ, ϑ ) g(u, u ) = (r 2 + a 2 )− 2 Qλ (ϑ, ϑ ) δ(u − u ) . (5.12) Note that if Qλ projects onto an eigenspace, N vanishes and thus g = s. Furthermore, since N is nilpotent, the series in (5.11) is actually a finite sum. Thus in view of Lemma 5.2, (5.11) is indeed well-defined. Proof of Lemma 5.3. Denoting the radial operator with integral kernel δ(u − u ) by 1u , we can write the result of Lemma 5.1 in the compact form (−∂u2 + V )s = 1u . Hence on the invariant subspace, we can do a Neumann series calculation, (−∂u2 + V ) g =
∞
(−N )l (−∂u2 + V )s l+1 =
l=0
∞
(−N )l s l = 1u − N g ,
k=0
to obtain that (−∂u2 + V + N ) g = 1u . According to (1.10), (1.7), and (5.2), this is equivalent to (5.12). We come to the separation of the resolvent. In order to explain the difficulty, we point out that H and Qλ do not in general commute, and thus (H − ω) Qλ = Qλ (H − ω)
and
Qλ S = S Qλ .
Therefore, one must be very careful with the orders of multiplication; in particular, it is not possible to simplify the operator product (H − ω)Qλ S. However, we know from the separation of variables that for every solution of the equation (H − ω) = 0, its projection Qλ (ω) is again a solution. In other words, H and Qλ do commute on the kernel of (H − ω). This fact will be exploited in the proof of the following proposition. Proposition 5.4. For ω ∈ σ (H ) we let Qλ be a spectral projector of the angular operator Aω . Then the resolvent of H has the representation Qλ S(ω) = Qλ T (ω, λ) , where T is the operator with integral kernel 00 T (u, ϑ; u , ϑ ) = δ(cos ϑ − cos ϑ ) δ(u − u ) 10 1 τ (u , ϑ ) σ (u , ϑ ) +δ(cos ϑ − cos ϑ ) (r 2 + a 2 )− 2 g(u, u ) . (5.13) ω τ (u , ϑ ) ω σ (u , ϑ )
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Here g is the Green’s function (5.11) (which depends on ω and λ), and τ, σ are the functions 3 σ = (r 2 + a 2 )− 2 (r 2 + a 2 )2 − a 2 sin2 ϑ , 3 τ = 2ak (r 2 + a 2 )− 2 (r 2 + a 2 ) − + ωδ . Proof. Let us compute the operator product (H −ω)Qλ T . We first consider the first summand in (5.13), which we denote by T1 . In this case, the operator product is particularly simple because the second column in the matrix (1.13) involves no u- or ϑ-derivatives. We obtain 1 0 ((H − ω) Qλ T1 )(u, ϑ; u , ϑ ) = Qλ (ϑ, ϑ ) δ(u, u ) . (5.14) β(u, ϑ) − ω 0 Next we consider the second summand in (5.13), which we denote by T2 . Fixing u , ϑ and considering T2 as a function of u, ϑ, we see from Lemma 5.3 that each column of Qλ T2 is for u < u a vector of the form = ( , ω ) with a solution of the separated wave equation (1.7). The same is true for u > u . Hence for u = u , Qλ T2 is composed of eigenfunctions of the Hamiltonian, ((H − ω) Qλ T2 )(u, ϑ; u , ϑ ) = 0
if u = u .
It remains to compute the distributional contribution to (H − ω)Qλ T2 at u = u . Since T2 is continuous at u = u , we only get a contribution when both radial derivatives act on the factor g. According to Lemma 5.2, the higher powers of s are in C 2 , and thus we may replace g by s. Applying (1.13), (2.18), and Lemma 5.1, we obtain ((H − ω) Qλ T2 )(u, ϑ; u , ϑ ) 1 0 0 = Qλ (ϑ, ϑ ) δ(u − u ) . τ (u , ϑ ) σ (u , ϑ ) σ (u, ϑ)
(5.15)
We add (5.14) to (5.15) and carry out the sum over λ ∈ σ (Aω ). Since the spectral projectors Qλ are complete (see Lemma 2.1 (iii)), λ Qλ (ϑ, ϑ ) gives a contribution only for ϑ = ϑ . We thus obtain a multiplication operator,
τ 00 (H − ω) Qλ T (λ) = 1 + (β − ω) + . σ 10
λ∈σ (Aω )
Using the explicit form of the functions τ , σ , and β, one sees that the second term vanishes. Thus (H − ω) Qλ T (λ) = 1 . λ∈σ (Aω )
Multiplying from the left by Qλ S and using the orthogonality of the angular spectral projectors gives the result.
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6. WKB Estimates In this section we shall derive estimates for the radial ODE (5.4) in the regime |Re ω| 1
and
|Im ω| ≤ c .
(6.1)
In this “high-energy regime”, the semi-classical WKB-solution should be a good approximation. In order to quantify this statement rigorously, we shall make an ansatz for φ which involves the WKB wave function and estimate √ the error. Our first lemma gives control of the sign of Re V . Lemma 6.1. There is a constant C such that for all ω with Im ω = 0 , |Re ω| > C √ and all λ ∈ σ (Aω ), the function Re V has no zeros. √ Proof. At a zero of Re V , the function V is real and non-positive. Thus it suffices to show that the imaginary part of V has no zeros. We first estimate the imaginary part of the angular spectrum. For any λ ∈ σ (Aω ) we let λ be a corresponding eigenvector. Then 1 1
, Aω L2 − Aω , L2 =
, (Aω − A∗ω ) L2 , Im (λ) , L2 = 2i 2i where ., .L2 is the L2 -scalar product on S 2 . Hence, according to (1.9), 1 1 ∗ 2 2 (aω sin ϑ + k) Aω − Aω = sup Im |Im λ| ≤ 2 2 sin ϑ 2 S
≤ 2a |Re ω| |Im ω| + |2ak Im ω| . 2
The imaginary part of (5.5) is computed to be ak Im ω + 2 Im λ Im V = −2 Re ω + 2 2 r +a (r + a 2 )2
(6.2)
(6.3)
Using (6.2), the second summand is estimated by |k| a2 (r 2 + a 2 )2 Im λ ≤ 2 (r 2 + a 2 )2 |Re ω| + a |Im ω| . The factor a 2 (r 2 + a 2 )−2 vanishes on the event horizon and at infinity and is always smaller than one. Thus there is a constant c with a 2 (r 2 + a 2 )−2 ≤ c < 1. This shows that after choosing |Re ω| sufficiently large, the first summand in (6.3) dominates the second, and so Im V has no zeros. In what follows, we assume that the assumptions of the above lemma are satisfied. We choose the sign convention for the square root such that 1 Re V (u), Re V (u) 4 ≥ 0 for all u ∈ R. (6.4) Furthermore, we shall restrict attention to ω in the range −c < Im ω < 0 ,
|Re ω| > C ,
(6.5)
where c is any fixed constant and C will be chosen depending on√the particular application. The next lemma, which we will need in Sect. 7, estimates V inside the “finite box” [uL , uR ] uniformly in ω for large Re ω.
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Lemma 6.2. For every angular momentum mode n and every c, ε > 0, there are constants C and c such that for all u ∈ [uL , uR ] and all ω in the range (6.5), √ (6.6) |Re V | ≤ c , (6.7) |Im V (ω) − Im V (Re ω)| ≤ ε . Proof. We set ω0 = Re ω, λ = λ(ω0 ) and introduce for a parameter τ ∈ [0, 1] the potential Vτ = V (ω0 ) + τ W with ak W = −2i Im ω Re ω + 2 + (Im ω)2 + (λ − λ0 ) 2 . r + a2 (r + a 2 )2 Then V0 = V (ω0 ) and V1 = V (ω). The mean value theorem yields that W , Re V (ω) − Re V (ω0 ) ≤ sup Re √ 2 Vτ τ ∈[0.1] W . V (ω) − Im V (ω ) ≤ sup Im √ Im 0 2 Vτ τ ∈[0.1]
(6.8) (6.9)
By √ choosing C sufficiently large, we can clearly arrange that V (ω0 ) < 0, and thus Re V (ω0 ) = 0. Furthermore, one sees immediately from the explicit formulas for V , W together with the estimate for the angular eigenvalue (2.7) that Re W = O(|Re ω|0 ) , Vτ − iRe ω = O(|Re ω|0 ) . Using this in (6.8) and (6.9) gives the claim.
Im W = O(|Re ω|1 ),
We introduce the WKB solutions α´ and α` by u √ − 41 − 41 α(u) ´ = c´ V exp V , α(u) ` = c` V exp − 0
u√
V
, (6.10)
0
where c´ and c` are some normalization constants. A straightforward calculation shows that these functions satisfy the Schr¨odinger equation 1 V V˜ = V − . (6.11) 4 V We can hope that α´ and α` are approximate solutions of the radial equation (5.4). In order to estimate the error, we first write (5.4) as a first order system, 0 1 φ . (6.12) with = = V 0 φ α = V˜ α
with
Next we make for the ansatz = A
with
A=
α´ α` α´ α`
(6.13)
and a 2-component complex function. A is the fundamental matrix of the ODE (6.11) and thus 0 1 A. (6.14) A = ˜ V 0
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Differentiating through the ansatz for (6.13) and using (6.12, 6.14), we obtain that 0 0 A = A . (6.15) V − V˜ 0 The determinant of A is a Wronskian and thus constant. A short computation using (6.10) shows that det A = −2c´c. ` Hence we can easily compute the inverse of A by Cramer’s rule, 1 α` −α` −1 A =− , 2c´c` −α´ α´ and multiplying (6.15) by A−1 gives = −
1 −α` α´ −α` 2 (V − V˜ ) . α´ 2 α´ α` 2c´c`
Finally, we put in the explicit formulas (6.11) and (6.10) to obtain the equation −1 −f −1 , (6.16) = W f 1 where W and f are the functions W =
f = exp 2
1 V , 8c´c` V 23
u√
V
.
(6.17)
0
We shall now derive an estimate for the solutions of the ODE (6.16). The main difficulty is that when the function f is very large or close to zero, the matrix in (6.16) has large norm, making it impossible to use simple Gronwall estimates. Instead, we can use that according to (6.4), the function u √ |f | = exp 2 Re V 0
is monotone. Theorem 6.3. Assume that the potential V in the Schr¨odinger equation (5.4) satisfies the conditions (6.4) and that the function W defined by (6.17) is in L1 (R). Then there is a solution of the system of ODEs (6.16) with boundary conditions 1 lim (u) = . 0 u→−∞ This solution satisfies for all u ∈ R the bounds u ≤ e4 W 1 W 1 , 1 (u) − exp − W −∞
| 2 (u)| ≤ e4 W 1 W 1 |f (u)| .
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Proof. We set ρ = |f | and introduce new functions a and b by a = 1
and
2 . ρ
b=
(6.18)
According to (6.16), they satisfy the following ODEs, a = −W a − W b +
ρ b, f
ρ f b = W a + Wb . ρ ρ
This gives rise to the following differential inequalities,
ρ d √ 1 aa = Re a a ≤ −|a| Re W + |b| W , du |a| f 1 ρ f |b| = Re b b ≤ −|b| + |a| W + |b| Re W . |b| ρ ρ
|a| =
Using that ρ is monotone and that |f | = ρ, we obtain the simple inequality (|a| + |b|) ≤ 2|W | (|a| + |b|) . Integrating this inequality from v to u, −∞ < v < u < ∞, gives the “Gronwall estimate” u (|a| + |b|)(u) ≤ (|a| + |b|)(v) exp 2 |W | ≤ (|a| + |b|)(v) e2 W 1 . (6.19) v
We now let (v) be the solution of (6.16) with boundary conditions (v) (v) = (1, 0). In order to estimate (v) , we rewrite (6.16) as e
e−
u v
u v
(v)
(v)
W
1 (u)
W
2 (u)
= −W e = W e−
u v
u v
(v)
W W
2 , f (v)
f 1 .
We integrate and use (6.19) to obtain the inequalities u u v W (v) 3 W 1 1 (u) − 1 ≤ e |W |, e v u u − v W (v) 3 W 1 3 W 1 (u) ≤ e ρ W ≤ e ρ(u) e 2 v
(6.20) u
|W | ,
(6.21)
v
where in the last step we used the monotonicity of ρ. The inequalities (6.20) and (6.21) yield that for every ε > 0, there is a u˜ such that for all v, v < u, ˜ the exponential on the left side of (6.20) and (6.21) are arbitrarily close to one, and the integrals on the right can be made arbitrarily small. Thus ˜ < ε. Due to the factor ρ(u) on the right of (6.21), we even know that |( (v) − (v ) )(u)| ) (v) (v ˜ < ε (with b according to (6.18)). Since (6.16) is linear, (v) − (v ) |(b − b )(u)| is also a solution. Applying (6.19) for this solution and choosing v = u, ˜ we obtain that ˜ This for all u > u, ˜ |( (v) − (v ) )(u)| < cε with a constant c being independent of u.
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shows that (v) (u) converges as v → −∞, and that the above estimates are still true for v = −∞. The theorem now follows from (6.20) and (6.21) if we set v = −∞ and pull out a u u factor of e −∞ W and e− −∞ W , respectively. The above theorem has two immediate consequences: First, it yields the existence of solutions φ´ and φ` which decay exponentially at minus and plus infinity, respectively. Second, it gives very good control of the global behavior of these solutions if |Re ω| is large. Corollary 6.4. For every angular momentum mode n and every ω with Im ω < 0, there are solutions φ´ and φ` of the Schr¨odinger equation (5.4) which satisfy the boundary conditions ´ = 1 = lim eiωu φ(u) ` . lim e−iωu φ(u) u→∞
u→−∞
´ because φ` is obtained in exactly the same way if one Proof. It suffices to construct φ, considers the ODEs backwards in u (i.e. after transforming the radial variable according to u → −u). We choose as in Theorem 6.3 and let φ´ = φ be the corresponding solution of the Schr¨odinger equation given by (6.13) and (6.12). Note that the corollary only makes a statement on the asymptotic behavior of φ´ as u → −∞, and thus the behavior of φ´ on any interval [u0 , ∞), u0 < 0 is irrelevant. Thus we may freely modify the potential V on any such interval. In particular, we can change the potential V on [u0 , ∞) such that it is constant √for large u. For any ε > 0, we choose u0 so small and modify V|[u0 ,∞) such that Re V ≥ 0 and W 1 < ε/3 (this is possible because V decays for large |u| at least at the rate ∼ |u|−3 ). Then Theorem 6.3 applies, and we obtain that | 1 − 1| ≤ ε ,
| 2 | ≤ ε |f | .
Using these bounds in (6.13), one sees that |φ/α´ − 1| < ε and thus, after choosing the normalization constants c´ and c` in (6.10) appropriately, ´ ≤ 1+ε lim sup |e−iωu φ|
and
u→−∞
Since ε is arbitrary, the result follows.
´ ≥ 1−ε. lim inf |e−iωu φ| u→−∞
Proposition 6.5. For every n and c, ε > 0, there is a constant C > 0 such that for all ω in the range (6.5), the solutions φ´ and φ` of Corollary 6.4 are close to the (suitably normalized) WKB wave functions α´ and α, ` (6.10), in the sense that for all u ∈ R, φ` φ´ φ` φ´ and − 1 + − 1 ≤ ε . − 1 + − 1 ≤ ε α` α´ α` α´ The reason why we need to choose C large is that the functions V /|V |3/2 and W must be sufficiently small. More specifically, one can choose C such that W 1 ≤
ε 3
and
|V | |V |
3 2
≤
1 . 3
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Proof of Proposition 6.5. Using (2.7) in (5.5), one sees that in the strip −2c < Im ω < 0, the potential V satisfies the bound |V (u) + ω2 | ≤ c1 (1 + |ω|) . On t he other hand, differentiating (5.5) and using (2.18) and (2.7), one sees that in this strip, |V (u)| + |V (u)| ≤ (1 + |ω|) g(u) , where g is a function which decays for large |u| at least at the rate ∼ u−2 . Putting these estimates for V and V into (6.17), one sees that by choosing C sufficiently large, we can arrange that for all ω in the range (6.5), W 1 ≤ ε/3. Theorem 6.3 yields that | 1 − 1| ≤ ε ,
| 2 | ≤ ε |f | .
(6.22)
Dividing the first row in (6.13) by α, ´ we obtain the identity φ´ α` = 1 + 2 , α´ α´ and using (6.22) gives φ´ α` − 1 ≤ ε 1 + |f | . α´ α´ From the second row in (6.13) we obtain similarly, φ´ α` − 1 ≤ ε 1 + |f | . α´ α´ Finally, we apply the elementary estimates for the WKB wave functions α` = 1 , α´ |f |
V V − 45 + V α` 1 = α´ |f | V V − 45 − V
2 ≤ , 1 |f | 4 1 4
where in the last step we applied the above bounds for V and V and possibly increased C. The solution φ` is obtained similarly if one considers the Schr¨odinger equation (5.4) backwards in u and repeats the above arguments. The next two propositions give estimates for composite expressions. Proposition 6.6. Under the assumptions of Proposition 6.5, w(φ, ` ´ φ) − 1 ≤ 4ε . w(α, ´ α) `
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Proof. Rewriting the Wronskian as ´ φ) ` = φ´ φ` − φ´ φ` = α´ α` w(φ,
´ ` φ´ φ` φ φ − α ´ α ` , α´ α` α´ α`
we can put in the estimate of Proposition 6.5 to obtain ´ ` ` + |α´ α` | . ´ α) ` ≤ 4ε |α´ α| w(φ, φ) − w(α,
(6.23)
Furthermore, a short explicit calculation using (6.10) shows that √ ` w(α, ´ α) ` = 2 V α´ α, 3 1 1 V (α´ α) ` =− ´ α), ` α´ α` = − V V − 2 w(α, 2 V 4 and thus
(6.24) (6.25)
3 1 1 1 w(α, ´ α) ` + (α´ α) ` = w(α, ´ α) ` 1 − V V −2 , 2 2 4 3 1 1 1 α´ α` = − w(α, ´ α) ` − (α´ α) ` = − w(α, ´ α) ` 1 + V V −2 . 2 2 4 α´ α` =
Substituting these relations into (6.23) gives w(φ, ` 1 − 3 ´ φ) 2 − 1 ≤ 4ε 1 + V V . w(α, ´ α) ` 4 Here the left side only involves Wronskians and is thus independent of u. Hence we may on the right side take the limit u → ∞. This gives the result. For uL < uR we set
´ L ) φ(u ` R ) − φ(u ´ R ) φ(u ` L ), φ[uL ,uR ] = φ(u
(6.26)
α[uL ,uR ] = α(u ´ L ) α(u ` R ) − α(u ´ R ) α(u ` L) . Proposition 6.7. Under the assumptions of Proposition 6.5, uR uR √ √ −1 φ[uL ,uR ] − 1 ≤ 8ε exp 2 Re V sin 2 Im V . α [uL ,uR ] uL uL Proof. Rewriting φ[uL ,uR ] as φ[uL ,uR ] = α(u ´ L ) α(u ` R)
´ L ) φ(u ´ R ) φ(u ` R) ` L) φ(u φ(u ` L) − α(u ´ R ) α(u , α(u ´ L ) α(u ` R) α(u ´ R ) α(u ` L)
Proposition 6.5 yields that φ[u ,u ] − α[u ,u ] ≤ 4ε |α(u ` R )| + |α(u ´ R ) α(u ` L )| ´ L ) α(u L R L R Furthermore, it is obvious from (6.10) that
` R ) = α(u ´ R ) α(u ` L ) exp −2 α(u ´ L ) α(u
uR
uL
√
V
(6.27)
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and thus from (6.26), uR √ α[uL ,uR ] = α(u ´ R ) α(u ` L ) exp −2 V −1 uL uR √ = α(u ´ L ) α(u ` R ) 1 − exp 2 V .
(6.28)
uL
Dividing (6.27) by α[uL ,uR ] and putting in the last identities, we obtain √ u 1 + exp −2 uLR V φ[uL ,uR ] 8ε ≤ , √ √ − 1 ≤ 4ε α u u [uL ,uR ] exp −2 uLR V − 1 exp −2 uLR V − 1 √ where in the last step we used that Re V ≥ 0. We finally estimate the obtained denominator from above, uR uR √ √ exp −2 V − 1 ≥ Im exp −2 V uL uL uR uR √ √ = exp −2 Re V sin 2 Im V . uL
uL
7. Contour Deformations In this section we shall use contour integral methods to prove the main theorem. Recall that in Sect. 3, we showed that the Hamiltonian HuL ,uR in finite volume is a selfadjoint operator on the Pontrjagin space PuL ,uR . It has a purely discrete spectrum, and for each ω ∈ σ (HuL ,uR ), the projector Eω onto the corresponding invariant subspace can be expressed as the contour integral
1 Eω = − Su ,u (ω ) dω , 2πi |ω −ω|<ε L R where ε is to be chosen so small that Bε (ω) contains no other points of the spectrum. The theory of Pontrjagin spaces also yields that σ (HuL ,uR ) will in general involve a finite number of non-real spectral points, which lie symmetrically around the real axis. We let EC be the projector onto the invariant subspace corresponding to all non-real spectral points, EC :=
Eω .
(7.1)
ω∈σ (HuL ,uR )\R
Our first lemma represents EC as a Cauchy integral over an unbounded contour. More precisely, we choose a contour CuL ,uR in the lower half plane which joins the points +∞ with −∞ and encloses the spectrum in the lower half plane from above. Furthermore, if Re ω is outside the finite interval [ω− , ω+ ], ω should be in the open set (see (4.6)) and should approach the real axis as |Im ω| ∼ −|Re ω|−1 (see Fig. 2).
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Fig. 2. Unbounded contour representation of EC
Lemma 7.1. The spectral projector corresponding to the non-real spectrum (7.1) has the representation EC =
1 lim Im π L→∞
CuL ,uR
L3 Su ,u (ω) dω . (L + iω)3 L R
(7.2)
Proof. The Cauchy integral formula yields that 1 EC = − 2π i
1 SuL ,uR (ω) dω + 2πi C
1 SuL ,uR (ω) dω = Im π C
C
SuL ,uR (ω) dω ,
where C is a closed contour which encloses the spectrum in the lower half plane (see Fig. 3). The dominated convergence theorem allows us to insert a factor L3 /(L + iω)3 , EC =
1 lim Im π L→∞
C
L3 Su ,u (ω) dω . (L + iω)3 L R
The function L3 /(L + iω)3 has no poles in the lower half plane and decays cubically for large |ω|. Furthermore, according to (4.9), the resolvent grows at most linearly for large |ω|. This allows us to deform the contour in such a way that C is closed in the lower half plane on larger and larger circles |ω| = R. In the limit R → ∞ the contribution along the circle tends to zero. Thus we end up with the integral along the contour CuL ,uR . Our next goal is to get rid of the “convergence generating factor” L3 /(L + iω)3 in (7.2). We shall use the fact that when we take the difference SuL ,uR − S∞ and evaluate it with a test function, the resulting expression has much better decay properties at infinity (see Lemma 4.3). We choose a contour C∞ which coincides with CuL ,uR if Re ω ∈ [ω− , ω+ ] and always stays inside (see Fig. 2). Lemma 7.2. For every ∈ C0∞ ((uL , uR ) × S 2 )2 , 1 < , EC >= Im π
CuL ,uR
< , SuL ,uR > dω −
C∞
< , S∞ > dω . (7.3)
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∂ C σ (HuL ,uR )
C ∂
Fig. 3. Closed contour representation of EC
Furthermore, < , EC > =
with
In
n∈IN
1 In = − 2πi
(7.4)
CuL ,uR
< , Qn SuL ,uR > dω −
1 + < , Qn SuL ,uR > dω 2πi CuL ,uR − < , Qn S∞ > dω .
C∞
< , Qn S∞ > dω
(7.5)
C∞
The series in (7.4) converges absolutely. We point out that the above integrals are merely a convenient notation and are to be given a rigorous meaning as follows. We formally rewrite the integrals in (7.3) (and similarly in (7.5)) as
C∞
< , (SuL ,uR − S∞ ) > dω +
− CuL ,uR
C∞
< , SuL ,uR > dω . (7.6)
Now the first summand is well-defined according to Lemma 4.3. In the second summand, the integrals combine to an integral over a bounded contour, and this is clearly well-defined because the contour does not intersect the spectrum of HuL ,uR . Note that in (7.5) we cannot combine the integrals over CuL ,uR and CuL ,uR (and similarly over C∞ and C∞ ) to the imaginary part of one contour integral because Qn in general does not commute with SuL ,uR , and so the integrands in (7.5) need not be real. For notational convenience, we abbreviate the second line in (7.5) by “−ccc” (for “complex conjugated contours”).
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Proof of Lemma 7.2. According to Corollary 4.2, the resolvent S∞ (ω) is holomorphic for ωin and grows at most linearly in |ω|. Thus for all L > 0, L3 S (ω) dω = 0 . 3 ∞ C∞ (L + iω) Combining this identity with (7.2), we obtain the representation < , EC > =
1 lim π L→∞ × Im − C∞
L3 < , SuL ,uR > dω 3 CuL ,uR (L + iω) L3 < , S > dω . ∞ (L + iω)3
We now rewrite the integrals according to (7.6). If we replace the contour CuL ,uR by C∞ , the integrands combine, and we obtain the expression 1 L3 lim Im < , (SuL ,uR − S∞ ) > dω . 3 π L→∞ C∞ (L + iω) The estimate (4.10) allows us to apply Lebesgue’s dominated converge theorem and to take the limit L → ∞ inside the integrand. The error we made when replacing CuL ,uR by C∞ is 1 L3 lim Im − < , SuL ,uR > dω . 3 π L→∞ CuL ,uR C∞ (L + iω) Now the contour is bounded, and since the factor < , SuL ,uR > is bounded, we can again apply Lebesgue’s dominated convergence theorem to take the limit L → ∞ inside the integrand. This gives (7.3). Note that our contours were chosen such that the condition (2.6) is satisfied for a suitable constant c > 0, and so Lemma 2.1 applies. Using completeness of the (Qn )n∈IN (see Lemma 2.1 (iii)), it immediately follows from (7.3) that < , EC > 1 =− < , Qn SuL ,uR > dω − < , Qn S∞ > dω 2π i CuL ,uR C∞ −ccc .
n∈IN
n∈IN
Again replacing the contour CuL ,uR by C∞ , we obtain the expression 1 − < , Qn (SuL ,uR − S∞ ) > dω − ccc . 2π i C∞ n∈IN
According to (4.11), the summands decay faster than any polynomial in λn . Applying the angular estimates (2.8) and (2.9), we conclude that the sum over n converges absolutely, uniformly in ω ∈ C∞ . Thus the dominated convergence theorem allows us to commute
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summation and integration, and the series converges absolutely. It remains to consider the expression 1 − − < , Qn SuL ,uR > dω − ccc . 2π i CuL ,uR C∞ n∈IN
Now the contours are compact, and thus the absolute convergence of the n-series is uniform on the contour. Hence we can again apply Lebesgue’s dominated convergence theorem to interchange the summation with the integration. We shall now deform the contours CuL ,uR and C∞ and analyze the resulting integrals. Our aim is to move the contours onto the real axis such that they reduce to an ω-integral over the real line. It is a major advantage of (7.4) that the series stands in front of the integrals, because this allows us to deform the contours in each summand In separately. Moreover, since our contour deformations will keep the values of the integrals unchanged, Lemma 7.2 guarantees that the series over n will converge absolutely. Thus we may in what follows restrict attention to fixed n. For given n, we know from Sect. 3 that the function < , Qn SuL ,uR > is meromorphic, and all poles are points of σ (HuL ,uR ). For the integrals over C∞ in (7.5), we cannot use abstract arguments because we have hardly any information on the spectrum of H∞ (we only know from Lemma 4.1 that the spectrum lies outside the set , (4.6), but it may be continuous and complex). But from the separation of the resolvent we know that ´ φ) ` vanishes the operator Qλ S∞ is well-defined and bounded unless the Wronskian w(φ, (see Proposition 5.4 and (5.7)). If this Wronskian were zero and Im ω < 0, this would give rise to a solution φ of the reduced wave equation which decays exponentially as u → ±∞. Such “unstable modes” were ruled out by Whiting [21]. We conclude that < , Qn S∞ > is analytic in the whole lower half plane {Im ω < 0}. Using the above analyticity properties of < , Qn SuL ,uR > and < , Qn S∞ >, we are free to deform the contours CuL ,uR and C∞ in any compact set, provided that CuL ,uR never intersects σn (HuL ,uR ). In particular, choosing ω− and ω+ real and outside of σ (HuL ,uR ), we may deform the contours as shown in Fig. 4. We let E[ω− ,ω+ ] be the projector on all invariant subspaces of HuL ,uR corresponding to real ω in the range ω− ≤ ω ≤ ω+ , Qn E[ω− ,ω+ ] = Qn (ω) Eω . ω∈[ω− ,ω+ ]
The next lemma shows that the integral over CI I I ∪ CI I I equals Qn E[ω− ,ω+ ] , whereas the integrals over the contours II and IV can be made arbitrarily small by choosing |ω± | sufficiently large.
Fig. 4. Contour deformation onto the real axis
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Lemma 7.3. For every ∈ C0∞ ((uL , uR )×S 2 )2 , n ∈ N, and ε > 0 there are ω− , ω+ ∈ R \ σ (HuL ,uR ) such that ≤ ε, In + < , Qn E[ω ,ω ] > − 1 < , Q − S > n ∞ − + 2πi CI CI where In are the integrals (7.5) and CI is any contour in the lower half plane which joins ω− with ω+ (see Fig. 4). Proof. Lemma 4.3 yields that by choosing ω+ and −ω− sufficiently large, we can make the contribution of the contour I V arbitrarily small. The integrals over CI I I and CI I I combine to contour integrals around the spectral points on the real axis, 1 < , Qn SuL ,uR > dω = − − 2πi CI I I CI I I
1 − < , Qn SuL ,uR > dω , 2πi |ω−ω |=δ ω
where the sum runs over all ω ∈ σ (HuL ,uR ) ∩ [ω− , ω+ ], and δ must be chosen so small that each contour contains only one point of the spectrum. If we let δ → 0 and use that Qn depends smoothly on ω, one sees that the integrals over the circles converge to −2π i < , Qn Eω >. We conclude that 1 − < , Qn SuL ,uR > dω = − < , Qn E[ω− ,ω+ ] > . − 2π i CI I I CI I I It remains to show that by choosing |ω± | sufficiently large, we can make the integral over the contour I I arbitrarily small. According to Lemma 2.1, for sufficiently large |ω± | the angular operator Aω is diagonalizable for all ω on the contour I I . Thus we can assume that the nilpotent matrices N in the Jordan decomposition (5.9) all vanish. Hence we can separate the resolvents according to Proposition 5.4 to obtain < , Qλ Tλ > , < , Qn (SuL ,uR − S∞ ) > = λ∈n
where Tλ is the operator with integral kernel 1
Tλ (u; u , ϑ ) = (r 2 + a 2 )− 2 (suL ,uR − s∞ )(u, u )
ρ(u , ϑ ) σ (u , ϑ ) . ω ρ(u , ϑ ) ω σ (u , ϑ )
Since the functions ρ and σ are smooth and the angular operators Qλ are bounded (2.9), it suffices to show that for every ε > 0 and g ∈ C0∞ ((uL , uR )), we can choose ω± such that for all ω on the contour I I , ∞ ∞ du du g(u) g(u ) (s[uL ,uR ] − s∞ )(u, u ) ≤ ε . −∞
−∞
Let us derive a convenient formula for s[uL ,uR ] −s∞ . We let φ1 and φ2 be the two fundamental solutions which satisfy the Dirichlet boundary conditions φ1 (uL ) = 0 = φ2 (uR ).
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Likewise, we let φ´ and φ` be the two fundamental solutions in infinite volume as constructed in Corollary 6.4. Furthermore, assume that uL < u < u < uR . Then, according to (5.7), suL ,uR (u, u ) =
1 φ1 (u) φ2 (u ) , w(φ1 , φ2 )
s∞ (u, u ) =
1 ´ ` ) . φ(u) φ(u ´ φ) ` w(φ,
Expressing φ1 as a linear combination of φ´ and φ2 , ´ ´ L ) φ2 (u) , φ2 (uL ) − φ(u φ1 (u) = φ(u) and substituting into the above formula for suL ,uR , we obtain 1 ´ ´ L ) φ2 (u) φ2 (u ) suL ,uR (u, u ) = φ(u) φ2 (uL ) − φ(u ´ φ2 ) φ2 (uL ) w(φ, ´ L ) φ2 (u) φ2 (u ) φ(u 1 ´ φ(u) φ2 (u ) − . = ´ φ2 ) ´ φ2 ) φ2 (uL ) w(φ, w(φ, ` In the first summand, we can express φ2 in terms of φ´ and φ, ´ ` R ) − φ(u ´ R ) φ(u) ` φ(u . φ2 (u) = φ(u)
(7.7)
This gives 1 1 ´ ´ ´ ) φ(u ` R ) − φ(u ´ R ) φ(u ` ) φ(u) φ2 (u ) = φ(u) φ(u ´ φ2 ) ´ R ) w(φ, ´ φ) ` w(φ, −φ(u ´ ´ ) ` R ) φ(u) φ(u φ(u + s∞ (u, u ) . =− ´ ` ´ φ(uR ) w(φ, φ) We conclude that (suL ,uR − s∞ )(u, u ) = −
` R ) φ(u) ´ ´ ) ´ L ) φ2 (u) φ2 (u ) φ(u φ(u φ(u − , ´ R ) w(φ, ´ φ2 ) ´ φ) ` φ2 (uL ) w(φ, φ(u
and because of its symmetry in u and u , this identity is also valid in the case u < u. Using (7.7) and the notation (6.26), we get (suL ,uR − s∞ )(u, u ) = We choose |ω± | such that uR uL
´ L ) φ2 (u) φ2 (u ) ´ ´ ) ` R ) φ(u) φ(u φ(u φ(u − . ´ R ) w(φ, ´ R ) w(φ, ´ φ) ` ´ φ) ` φ[uL ,uR ] φ(u φ(u
(7.8)
2Z + 1 π. Im V (ω± ) ∈ 4
According to the estimate (6.7) in Lemma 6.2, we can arrange that the function √ V is nearly constant on the contour II, and thus u R √ 1 sin 2 Im V ≥ . 2 uL
uR uL
Im
(7.9)
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Propositions 6.5, 6.6, and 6.7 allow us to estimate each term in (7.8) by the corresponding term in the WKB approximation. According to (7.9), the factor | sin(. . . )|−1 which appears in Proposition 6.7 is bounded. Choosing ε sufficiently small, we thus obtain the estimate (su ,u − s∞ )(u, u ) L R uR √ α(u ´ L ) α2 (u) α2 (u ) ≤ 2 Re V exp 2 α[uL ,uR ] α(u ´ R ) w(α, ´ α) ` uL α(u ´ α(u ´ ) ` R ) α(u) , (7.10) +2 α(u ´ R ) w(α, ´ α) ` where we introduced the function α2 (u) = |α(u) ´ α(u ` R )| + |α(u ´ R ) α(u)| ` . Using the explicit formulas (6.24, 6.28) together with (7.9), we get |w(α, ´ α)| ` ≥ | V (u)| |α(u) ´ α(u)| ` ,
|α[uL ,uR ] | ≥
1 ` R )| . (7.11) |α(u ´ R ) α(u 2
Substituting these bounds into (7.10), we get an estimate for |suL ,uR − s∞ | in terms of expressions of the form uR √ α(u α(u 1 ´ 1 ) ` k) · · · Re V · · · (7.12) exp 2 √ α(v ´ 1) α(v ` k) |V (u)| uL with ui , vi ∈ [uL , uR ]. The quotients of the WKB wave functions have according to (6.10) the explicit form − 1 α(u) 4 ´ = V (u) exp 2 α(v) V (v) ´
uR
√
Re V
(7.13)
,
uL
and similarly for α. ` The inequality (6.6) shows that the exponentials in (7.12) and (7.13) are bounded uniformly in ω. Furthermore, it is obvious from (5.5) that V (u)/V (v) is close to one if |ω| is large. We conclude that on the contour II, |(suL ,uR − s∞ )(u, u )| ≤ √
c , |V (u)|
and this can be made arbitrarily small by choosing |ω± | sufficiently large.
We are now in the position to prove our main theorem. Proof of Theorem 1.1. According to Lemma 7.3, 1 − < , Qn S∞ > dω = In + < , Qn ER > , lim − 2π i ω± →±∞ CI CI where ER denotes the projector onto the invariant subspace corresponding to the real spectrum of HuL ,uR . Here the ω± are to be chosen as in Lemma 7.3 and CI is again any contour which joins ω− with ω+ in the lower half plane. Suppose that the contour Cε
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intersects the lines Re ω = ω± in the points ω+ − iδ+ and ω− − iδ− , respectively. Then we choose the contour CI as follows, CI = (ω− − i[0, δ− ]) ∪ (Cε ∩ (ω− , ω+ ) + iR) ∪ (ω+ − i[0, δ+ ]) . The first and last parts of the contour have lengths δ− and δ+ , respectively, and these lengths clearly tend to zero as ω± → ±∞. Furthermore, it is obvious from Propositions 6.5 and 6.6 as well as (7.11) and (7.13) that the integrand is uniformly bounded on these parts of the contour. Hence the contribution of these contours tends to zero as ω± → ±∞. Thus 1 − < , Qn S∞ > dω = In + < , Qn ER > . − 2π i Cε Cε According to Lemma 7.2 and Lemma 2.1, the right side of this equation is absolutely summable in n and (In + < , Qn ER >) = < , (EC + ER ) > . n
Since the spectral projectors in the Pontrjagin space HuL ,uR are complete, EC +ER = 1. We conclude that 1 − < , Qn S∞ > dω = < , > . − 2π i n Cε Cε Polarizing, we obtain for every ∈ C0∞ ((r1 , ∞) × S 2 )2 the simple identity 1 Qn S∞ dω . − = − 2πi n Cε Cε
(7.14)
The integral and sum converge in L2loc . If we apply the Hamiltonian to the integrand in the above formula, we obtain according to Proposition 5.4, H Qn S∞ = (H − ω) Qn S∞ + ω Qn S∞ = (holomorphic terms) + ω Qn S∞ . The holomorphic terms are holomorphic in the whole neighborhood of the real axis enclosed by Cε and Cε (see Lemma 2.1 (i)), and therefore the contour integral over them drops out. We conclude that applying H reduces to multiplying the integrand by a factor ω. Iteration shows that the dynamics of is taken into account by a factor e−iωt , 1 (t) = − e−iωt Qn S∞ 0 dω . − 2πi n Cε Cε Comparing this expansion with (7.14), one sees that the integrand in the last expansion is equal to the integrand in (7.14) if is replaced by (t). Since (t) is smooth and by causality has compact support, we conclude that the integral and sum again converge in L2loc . Finally, using that the contour integrals in this formula are all independent of ε, we may take the limit ε 0 of each of them. Acknowledgements. We are grateful to Harald Schmid and Johann Kronthalar for helpful discussions. We would like to thank McGill University, Montr´eal, the Max Planck Institute for Mathematics in the Sciences, Leipzig, and the University of Michigan for support and hospitality. We are also grateful to the Vielberth Foundation, Regensburg, for generous support.
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References 1. Anco, S.: Private communication 2. Beyer, H.: On the stability of the Kerr metric. Commun. Math. Phys. 221(3), 659–676 (2001) 3. Bognar, J.: Indefinite Inner Product Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78, New York-Heidelberg: Springer-Verlag, 1974 4. Carter, B.: Black hole equilibrium states. In Black holes/Les astres occlus, Ecole d’ e´ t´e Phys. Th´eor., Les Houches, 1972 5. Chernoff, P.: Self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12, 401–414 (1973) 6. Coddington, E., Levinson, N.: Theory of Ordinary Differential Equations. NewYork: Mc Graw-Hill, 1955 7. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics 19, Providence, RI: American Mathematical Society, 1998 8. Finster, F., Kamran, N., Smoller, J., Yau, S-T.: The long-time dynamics of Dirac particles in the Kerr-Newman black hole geometry. Adv. Theor. Math. Phys. 7, 25–52 (2003) 9. Finster, F., Kamran, N., Smoller, J., Yau, S-T.: Decay of solutions of the wave equation in the Kerr geometry. http://arxiv.org/abs/gr-qc/0504047, 2005 10. Finster, F., Schmid, H.: Spectral estimates and non-selfadjoint perturbations of spheroidal wave operators. http://arxiv.org/abs/math-ph/0405010, 2004 11. Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. BerlinHeidelberg-New York: Springer-Verlag, 1998 12. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press, 1973 13. Kato, T.: Perturbation Theory for Linear Operators. 2nd edition, Berlin-Heidelberg-New York: Springer, 1995 14. Kay, B., Wald, R.: Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere. Classical Quantum Gravity 4(4), 893–898 (1987) 15. Klainerman, S., Machedon, M., Stalker, J.: Decay of solutions to the wave equation on a spherically symmetric background. Preprint (2002) 16. Langer, H.: Spectral functions of definitizable operators in Krein spaces. In: Functional analysis (Dubrovnik, 1981), Lecture Notes in Math. 948, Berlin-New York: Springer, 1982, pp. 1–46 17. Leray, J.: Hyperbolic differential equations. Princeton, NJ: The Institute for Advanced Study, 1953, 238 pp. Reprinted November 1955 18. Nicolas, J.-P.: A nonlinear Klein-Gordon equation on Kerr metrics. J. Math. Pures Appl. 81, 885–914 (2002) 19. Taylor, M.: Partial differential equations I. Berlin-Heidelberg-New York: Springer-Verlag, 1997 20. Taylor, M.: Partial differential equations I. Berlin-Heidelberg-New York: Springer-Verlag, 1997 21. Whiting, B.: Mode stability of the Kerr black hole. J. Math. Phys. 30, 1301–1305 (1989) Communicated by G.W. Gibbons
Commun. Math. Phys. 260, 299–317 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1404-8
Communications in
Mathematical Physics
Gravitational Instantons of Type Dk Sergey A. Cherkis1,2 , Nigel J. Hitchin3 1 2 3
Institute for Advanced Study, Princeton, NJ 08540, USA. E-mail: [email protected] Trinity College, Dublin 2, Ireland Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK. E-mail: [email protected]
Received: 27 August 2004 / Accepted: 22 February 2005 Published online: 31 August 2005 – © Springer-Verlag 2005
Abstract: We use two different methods to obtain Asymptotically Locally Flat hyperk¨ahler metrics of type Dk . 1. Introduction A self-dual gravitational instanton is a hyperk¨ahler manifold of real dimension four. These can be distinguished from each other by the asymptotic behaviour of the metric and their topology. We give in this paper explicit formulas for asymptotically locally flat (ALF) hyperk¨ahler metrics of type Dk . The Ak case has been known for a long time as the multi-Taub-NUT metric of Hawking [1] and is given in terms of one harmonic function V (x) on R3 , V (x) = 1/µ +
k+1
1/|x − xi |.
1
The metric is ds 2 = V dx · dx + V −1 (dθ + ω)2 ,
(1)
where dω = ∗3 dV and ω is to be interpreted as a connection form for a principal circle bundle over the points of R3 where V is non-singular. Over a large 2-sphere, this bundle has Chern class k + 1. Now V ∼ 1/µ + (k + 1)/|x| is invariant under x → −x and an ALF metric of type Dk is asymptotic to the quotient of an A2k−5 metric by the action of this reflection. There are also degenerate cases within the family – the D0 case is the 2-monopole moduli space calculated in [2] and the D2 case was highlighted in [3] as an approximation to the K3 metric. Both the derivation and formula for the D0 metric benefited from the presence of continuous symmetry groups whereas, as we show here,
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the general Dk case has none, which perhaps explains the more complicated features of what follows. The actual manifold on which the metric is defined is, following [4], most conveniently taken to be a hyperk¨ahler quotient by a circle action on the moduli space of U (2) monopoles of charge 2 with singularities at the k points q1 , . . . , qk ∈ R3 . The original interest in physics of self-dual gravitational instantons (of which these metrics are examples) was motivated by their appearance in the late seventies in the formulation of Euclidean quantum gravity [5, 1, 6]. In this context they play a role similar to that of self-dual Yang-Mills solutions in quantum gauge theories [7]. Since then, however, these objects have appeared in various problems of quantum gauge theory, string theory and M-theory, some of which we now mention. For concreteness, we concentrate on the case of the Dk -type ALF gravitational instanton: • compactifications of supergravity, string and M-theory on self-dual gravitational instantons preserve half the amount of supersymmetry of the original theory – M-theory on Dk ALF spaces, for example, emerges as a strong string coupling limit of type IIA string theory with an O6-plane in the presence of k D6-branes [8] • as discussed by Seiberg and Witten [9], quantum moduli spaces of supersymmetric N = 4 SU (2) gauge theories with k fundamental hypermultiplets in three dimensions are Dk ALF spaces • these spaces can also be considered as moduli spaces of solutions of Bogomolny equations with prescribed singularities, or as moduli spaces of instantons on multiTaub-NUT spaces that are invariant with respect to the S 1 symmetry [10]. We derive the formulas by two different methods. The first is based on unpublished work of the second author carried out for the ALE case during a visit to the University of Bonn in 1979. It followed the twistor approach to the Ak case in [11], except that one needed a polynomial solution to x 2 − zy 2 = a (“Pell’s equation"), where z(ζ ) is a quartic, instead of the simple factorization xy = a as in the Ak case. This was solved by introducing the elliptic curve w2 = z(ζ ) and trying to factorize a = (x − wy)(x + wy) into elliptic functions. There is a divisor class constraint (considered in 1828 by Abel [12]!) to doing this. Given the more recent interpretation of the ALF solutions in terms of monopoles, the analogous elliptic curve is naturally described as the spectral curve of the monopole, or the equivalent Nahm data. As with all spectral curves, it is subject to a transcendental constraint and the key problem in writing down the metric is to implement analytically this constraint on z(ζ ), cutting down the five coefficients of the quartic to give four coordinates on the hyperk¨ahler manifold. The second method uses the generalized Legendre transform construction of Lindstr¨om, Ivanov and Roˇcek [13, 14] which has already been successfully used for k = 0 and has begun to be applied to the problem considered here by the first author in [15]. In this case the quartic z(ζ ) appears in a fundamental way and the constraint is expressed by a differential equation. We show how these two expressions for the constraint coincide. When the singularities of the monopole lie on a line through the origin, the metric defines an explicit resolution of the Dk quotient singularity by a configuration of holomorphic 2-spheres, intersecting according to the Dk Dynkin diagram. We hope to consider this special case, where there does exist a circle action, in more detail elsewhere. 2. Singular Monopoles We shall use certain moduli spaces of singular monopoles to obtain our Dk metrics, as in [15]. There are two approaches to this: through Nahm’s equations, which give us
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a concrete analytical description of the objects in the space and the L2 metric defined on it, and the twistor approach which we use here. The latter describes a generic point in the moduli space via an algebraic curve, and allows us the possibility of an explicit determination of the metric. 2.1. The twistor approach to monopoles. It was shown in [4], using the technique developed in [16] as well as results of Kronheimer [17], that a charge 2 U (2) monopole solution to the Bogomolny equations with k singularities is described by a spectral curve S in T P1 and two sections π and ρ of holomorphic line bundles L−µ (k)|S and Lµ (k)|S respectively. Viewing T P1 as the total space of O(2) we denote by ζ an affine coordinate on P1 and η the standard linear coordinate in the fibre. Then Lµ (k) → T P1 denotes the line bundle with transition function ζ −k exp(µη/ζ ). There is a real structure σ given by σ (η, ζ ) = (−η, ¯ −1/ζ¯ ). The spectral curve S ⊂ T P1 for a charge 2 monopole is then given by an equation η2 − yη − z = 0,
(2)
where y is a real section of O(2) and z is a real section of O(4). The reality condition for a section x of O(2n) is x(σ (ζ )) = (−1)n x(ζ )/ζ 2n . Thus in the patch ζ = ∞ the section x is given by a polynomial x(ζ ) of degree 2n which satisfies x(−1/ζ¯ ) = (−1)n x(ζ )/ζ 2n . The position of each of the k singularities of the monopole configuration can be described by a real section qi of O(2), i = 1, . . . , k. We denote these sections by Pq1i ⊂ T P1 . The two sections π and ρ on the spectral curve are interchanged by the real structure and satisfy πρ =
k
(η − qi ) .
(3)
i=1
There is a circle action on this data given by (ρ, π ) → (λρ, λ−1 π ).
(4)
Equation (3) says that the intersection of S with all the curves Pq1i defines (if S does not contain one of them as a component) a divisor on S of degree 4k, and the constraint on the spectral curve is expressed by the fact that we can divide these into two sets of 2k points, one of which is a divisor for L−µ (k)|S and the other for Lµ (k)|S . 2.2. The twistor approach to the moduli space. The spectral curve description fits into the twistor description of the hyperk¨ahler metric on the moduli space, which we recall from [4], and is similar to the case of non-singular monopoles in [2]. Let D = {(η, y, z) ∈ O(2) ⊕ O(2) ⊕ O(4)|η2 − yη − z = 0} which has a projection p1 (η, y, z) = η onto T P1 and another p2 (η, y, z) = (y, z) which represents D as a ramified double covering of O(2) ⊕ O(4). We let V µ be the rank 2 holomorphic vector bundle on O(2) ⊕ O(4) which is the direct image sheaf V µ = (p2 )∗ (p1∗ Lµ ).
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Then the direct image of Eq. (3) defines a subvariety Z of V µ (k) ⊕ V −µ (k) which will be a model of our twistor space. There is some resolution of singularities to be carried out, but that doesn’t affect the determination of the metric. In the twistor space approach we need to find the twistor lines, which are sections of p : Z → P1 . The quadratic and quartic y, z define a section of O(2) ⊕ O(4) and the functorial property of the direct image says that π, ρ define a lifting to Z ⊂ V µ (k) ⊕ V −µ (k). Over ζ = 0, the direct image equation for Z can be written as (x1 + ηx2 )(y1 + ηy2 ) = (η − qi ) mod η2 − yη − z = 0 i
or equivalently, x1 y1 + zx2 y2 = p(y, z), x2 y1 + x1 y2 + yx2 y2 = q(y, z), where
(5)
(η − qi ) = p + ηq mod η2 − yη − z = 0. i
This equation defines a 5-dimensional twistor space for an 8-dimensional hyperk¨ahler manifold – the moduli space of charge 2 singular monopoles. There is a symplectic form along the fibres which can be written as in [15] as ω=4
2 dρ(βj ) ∧ dβj j =1
ρ(βj )
,
(6)
with η = βj being roots of η2 − yη − z = 0, but in coordinates x1 , x2 , y, z above as ω=4
(x1 dx1 + (yx1 − zx2 )dx2 ) ∧ dy + (x1 dx2 − x2 dx1 ) ∧ dz . x12 − zx22 + yx1 x2
2.3. The hyperk¨ahler quotient. To obtain a 4-dimensional manifold we shall take a hyperk¨ahler quotient by a circle action which at the twistor space level is given by (4) and in the above coordinates is (x1 , x2 , y1 , y2 , y, z) → (λx1 , λx2 , λ−1 y1 , λ−1 y2 , y, z). In coordinates x1 , x2 , y, z the vector field generated by this action is X = x1
∂ ∂ + x2 ∂x1 ∂x2
so that i(X)ω = 4dy. Remark 1. The moment map for this action is 4y. From the point of view of monopoles this can be interpreted in terms of the centre of mass, in which case the more natural value would be y/2 which amounts to a rescaling of ω.
Gravitational Instantons of Type Dk
303
The hyperk¨ahler quotient from the twistor point of view is just the fibrewise symplectic quotient, so we set the moment map y = 0 (this means that the centre of mass of the monopole is at the origin) and take the quotient by the C∗ action. Putting y = 0 in (5), gives x1 y1 + zx2 y2 = p(y, z), x2 y1 + x1 y2 = q(y, z).
(7)
We need a smooth 4-manifold so the circle action on the zero set of the hyperk¨ahler moment map must be free. In the nonsingular monopole case this is automatic: up to a finite covering the moduli space is isometric to a product Mk0 × S 1 × R3 . The action is not necessarily free in the singular case. For example, the moduli space of charge 1 monopoles with k singularities is the Ak−1 ALF space – multi-Taub-NUT space – and the triholomorphic circle action has fixed points. Suppose we have a fixed point, then there is a fixed twistor line, so on each fibre of the twistor space a fixed point, which from (4) is where x1 = x2 = 0. This means p = q = 0. But with y = 0, p and q are defined by r(η) = (η − qi ) = p + ηq mod η2 − z = 0, i
thus p(z) =
1 (r(η) + r(−η)), 2
q(z) =
1 (r(η) − r(−η)), 2η
and p and q have a common zero if r(η) and r(−η) have a common zero, i.e. qi = −qj (for all ζ ). If qi = 0, then q(0) = r (0) so since the qi are distinct, the action doesn’t have a fixed point. Thus, so long as the positions of the singularities of the monopole satisfy qi = −qj for any i = j , we shall produce a non-singular 4-manifold as a hyperk¨ahler quotient. The equation of the twistor space of the quotient can be obtained by using the C∗ invariant coordinates P = 2x1 y1 − p,
Q = 2x2 y2 − q,
and then from (7) P −p Q+q x2 , =z =z x1 P +p Q−q which gives P 2 − zQ2 =
k
z − qi2 .
(8)
i=1
The holomorphic symplectic form along the fibres is then √ 1 P + zQ ω = d √ log ∧ dz. √ z P − zQ
(9)
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S.A. Cherkis, N.J. Hitchin
Remark 2. Equation (8) defines a subvariety of the rank 2 direct image vector bundle V 2µ (2k) over O(4). Concretely, V 2µ has transition matrix √ √ √ z sinh(2µ z/ζ ) cosh(2µ z/ζ ) P P˜ √ √ √ . ˜ = ζ 2 sinh(2µ z/ζ )/ z ζ 2 cosh(2µ z/ζ ) Q Q Note that if z˜ = z/ζ 4 then ˜ 2 = P 2 − zQ2 , P˜ 2 − z˜ Q which describes a global holomorphic quadratic √ form (u, u) on V , singular on the zero section z = 0. The null-directions are P ± zQ which are globally defined on the double covering O(2) and are the line bundles L±2µ . This quadratic form gives a more invariant way of writing (8): (u, u) =
k
z − qi2 .
(10)
i=1
2.4. The Dk singularity. The fibre over ζ ∈ P1 in the 3-dimensional variety defined by (10) has the equation P 2 − zQ2 =
k
z + pi2
(11)
i=1
if we set pi = iqi (ζ ). The universal deformation of the Dk singularity has the form k k k 1 2 2 2 2 x − zy = (z + pi ) − pi + 2i pi y, z i=1
i=1
i=1
and putting P = iyz −
k
pi ,
Q = ix,
i=1
we obtain (11). If the pi2 are distinct for a fixed ζ and if none of the pi vanish, (11) defines a smooth surface and (x, y, z) → (P , Q, z) is a biholomorphic map. At the finite number of values of ζ at which a pi vanishes, the fibre is singular and the actual twistor space Z for the hyperk¨ahler metric involves a resolution of these singularities. The singular points are replaced by compact rational curves, but since a generic twistor line misses the singularities, we do not need to use this fact to calculate the metric. On the other hand, our formulae are not sufficiently manageable here to use the explicit form of the metric to describe these resolutions, as was done in the Ak case in [11]. The existence of these compact curves in certain fibres gives us information about the Killing fields for the metric. Any such vector field induces a holomorphic vector field on the twistor space, taking fibres to fibres. The generic fibre is an affine surface in C3 and so has no compact subvarieties other than points, since the coordinate functions
Gravitational Instantons of Type Dk
305
must be constant. Thus the vector field must preserve those fibres with rational curves. There are then two alternatives. One is that it preserves all fibres and therefore defines a triholomorphic Killing field, but this is impossible: if an ALF space has a trihilomorphic isometry, then its metric is exactly equal to the metric given in Eq. (1). The other, since a holomorphic vector field on the sphere has at most two zeros, is that there are only two singular fibres, which is the case when all the pi vanish at two antipodal points on P1 . In this case the singular fibre is the Dk singularity x 2 − zy 2 = zk−1 . In another paper we shall explore more explicitly the resolution of this singularity by our twistor lines, but for the present this argument shows that our metrics for a generic choice of pi have no Killing fields. 3. The Equations Defining the Constraint The main problem in determining a metric in the twistor approach is to find the twistor lines – holomorphic sections of the projection Z → O(4) → P1 . From the construction above, we have a rational curve C in O(4) defined by z = z(ζ ), where z(ζ ) is a quartic polynomial, and to lift it further we needed a section u(ζ ) of V 2µ (2k) over C, which √ from the functorial property of the direct image was equivalent to a section a = P + zQ of For the section √ u(ζ ) to satisfy (10) is equivalent to the L2µ (2k) on the spectral curve. √ existence of sections a = P + zQ, b = P − zQ on the spectral curve S satisfying ab =
k
z − qi2 .
i=1
The spectral curve of the monopole had sections π, ρ such that πρ =
k
(η − qi ) .
i=1
Setting a(η, ζ ) = ρ(η, ζ )π(−η, ζ ),
b(η, ζ ) = (−1)k ρ(−η, ζ )π(η, ζ )
gives sections a, b of L2µ (2k)|S and L−2µ (2k)|S respectively such that ab =
k
z − qi2
i=1
is satisfied. If the αij are roots of z(ζ ) − qi2 (ζ ), then a vanishes at points (η, ζ ) =
1 j (−1)j qi (α
ij ), αij ∈ S ∈ T P . The section b vanishes at (η, ζ ) = −(−1) qi (αij ), αij . A rotation of R3 acts as a fractional linear transformation on the coordinate ζ . Using such a transformation we can put z(ζ ) in the form r1 ζ 3 − r2 ζ 2 − r1 ζ with real r2 and r1 ≥ 0. In these coordinates we shall solve the constraint equation for the spectral curve √ η2 = z(ζ ) using Weierstrass elliptic functions: η = r1 P (u)/2 and ζ = P(u)+r2 /3r1 , where u is the affine coordinate on C/ representing the torus η2 = z(ζ ). Note that η → −η corresponds to u → −u. In what follows we order the points αij so that ρ vanishes at αi2 and αi4 , while π vanishes at αi1 and αi3 . Let uij be the zeros of the
306
S.A. Cherkis, N.J. Hitchin
sections ρ and π corresponding to qi (αij ), αij , then the condition for the sections to be doubly periodic translates into √ 2µ r1 + (−1)j uij = 0, k
4
(12)
i=1 j =1
or in terms of z and ζ , (−1)j
αij
ij
dζ √ = −4µ. z
(13)
From the fact that a/b is a section of L4µ |S with zeroes at (−1)j uij and poles at √ √ −(−1)j uij it follows (see [4] and [15]) that with a = P + zQ, b = P − zQ and ζW the Weierstrass zeta function log
σ (u − (−1)j uij ) √ a = −2µ r1 (ζW (u + u∞ ) + ζW (u − u∞ )) + log . (14) b σ (u + (−1)j uij ) ij
Returning to z and ζ 1 a (−1)j log = √ b z(ζ ) ij
αij
dξ . √ (ξ − ζ ) z(ξ )
(15)
4. From the Twistor Space to the Metric The first calculation uses Penrose’s original nonlinear graviton construction [18]. First we calculate the conformal structure and then use the holomorphic form to determine the volume form. The 4-dimensional spacetime M is the space of twistor lines, a tangent vector is a holomorphic section of the normal bundle and it is a null vector for the conformal structure if and only if that section vanishes somewhere on the twistor line. This is the infinitesimal version of the statement that two points in M are null separated if the twistor lines meet.
4.1. The complex conformal structure. The twistor line is given by η2 = z(ζ ) = A
4
ζ − aj
(16)
j =1
and we define
√ a 1 P (ζ ) + z(ζ )Q(ζ ) 1 χ (ζ ) = √ log = √ log , √ z b z P (ζ ) − z(ζ )Q(ζ )
(17)
so that the symplectic form given by Eq. (9) is ω = dχ ∧ dz.
(18)
Gravitational Instantons of Type Dk
307
We can think of the variables aj and A satisfying the constraint (13) as providing four real coordinates on the Dk ALF manifold M. Later we shall change coordinates to z(0) and Q(0). Varying the parameters A and aj under the constraint (13) we are seeking the condition for an infinitesimal variation to vanish. From now on we denote the infinitesimal variations by a prime, e.g. A , aj , z , etc. Putting the variation of Eq. (16) to zero we have 4 z A aj = 0, = − z A ζ − aj
(19)
j =1
while the vanishing of the variation of the constraint (13) gives αij αij dζ A al j j = , (−1) (−1) − √ ζ − al 2 z(ζ ) A z(αij ) ij
ij
(20)
l
and for the second term on the right hand side we note that αij 1 dζ 1 j (−1) = χ (al ) = √ 2 2 (ζ − a ) z(ζ ) l ij √ 1 P (ζ ) + z(ζ )Q(ζ ) Q(al ) , = lim √ log = √ ζ →al 2 z(ζ ) P (al ) P (ζ ) − z(ζ )Q(ζ )
(21)
as z(al ) = 0. Let us introduce a function L(ζ ) =
i
li (ζ ) , z(ζ ) − qi2 (ζ )
(22)
where li are cubic polynomials in ζ such that li (αij ) = qi (αij ) (Lagrange interpolation polynomials). Also let l = lim ζ L(ζ ). ζ →∞
(23)
Then using the constraint equation (13) and substituting αij in Eq. (20) we find 4 A Q(al ) al L(al ) − . (2µ − l) = A P (al )
(24)
4 al 1 z Q(al ) χ = L(ζ ) − χ (ζ ) + . L(al ) − 2 z P (al ) ζ − al
(25)
l=1
The variation of χ is
l=1
Requiring the variation of χ and z to vanish gives firstly 4 l=1
al Q(al ) = 0, L(al ) − P (al ) ζ − al
(26)
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S.A. Cherkis, N.J. Hitchin
and secondly, eliminating A from Eqs. (19) and (24) leads to 4 l=1
al Q(al ) 2µ − l + al L(al ) − = 0. ζ − al P (al )
(27)
Equations (26) and (27) give a system of two equations cubic in ζ and linear in al . Our task is to find the condition on the coefficients of these equations to have a common solution for ζ. This condition is provided by the vanishing of the resultant R of the system (26, 27), which is a polynomial in al of degree six. However, if aj = 0 then ζ = aj solves the equations and so R must be divisible by a1 a2 a3 a4 . This means S(al ) = R(al )/(a1 a2 a3 a4 )
(28)
is a polynomial which is quadratic in the variations and so is the quadratic form defining the conformal structure. For convenience we introduce Ci = L(ai ) −
Q(ai ) 1 = L(ai ) − χ (ai ), P (ai ) 2
Di = 2µ − l + ai Ci .
(29)
Computing the resultant (see the Appendix) we find 4 0 i=1 Di dai 0 0 4 0 − i=1 Di dai /ai S(dai ) = 4 C da 0 i=1 i i 0 0 0 − 4i=1 Ci dai /ai
a12 D1 a 1 D1 D1 a12 C1 a 1 C1 C1
a22 D2 a2 D2 D2 a22 C2 a2 C2 C2
a32 D3 a3 D3 D3 a32 C3 a3 C3 C3
a42 D4 a4 D4 D4 , a42 C4 a4 C 4 C4
(30)
where we use differentials dai instead of the primes ai . 4.2. The real metric. To impose the reality condition, we note that the map ζ → −1/ζ¯ takes the set of four roots {a1 , a2 , a3 , a4 } into itself, so that for a permutation σ : (1, 2, 3, 4) → (2, 1, 4, 3) we have aσ (i) = −1/a¯i . Now noting that 1 L(− ) = ζ 2 L(ζ ) − lζ, ζ¯
(31)
1 χ (− ) = ζ 2 χ (ζ ) − 4µζ, ζ¯
(32)
Cσ (i) = ai Di , Dσ (i) = −ai Ci ,
(33)
aσ (i) Cσ (i) = −Di , aσ (i) Dσ (i) = Ci , Di Ci aσ2 (i) Cσ (i) = , aσ2 (i) Dσ (i) = − . ai ai
(34)
it follows that
(35)
Gravitational Instantons of Type Dk
309
With these relations, one can easily check that S¯ = S/(a1 a2 a3 a4 ), and the quantities A0 =
4
B0 =
Ci dai ,
i=1
4
(36)
Di dai
i=1
satisfy A0 =
4 i=1
Di
dai , ai
B0 = −
4 i=1
Ci
dai . ai
(37)
Thus, we find that the metric is proportional to the real symmetric two-form 1 A0 , S = (A0 B0 ) G √ B0 a1 a2 a3 a4 with 1 G= √ a1 a2 a3 a4
[a 2 D, aD, aC, C] −[a 2 D, aD, D, aC] . [aD, a 2 C, aC, C] [aD, D, a 2 C, aC]
(38)
(39)
Here we use the notation
e1 e2 [e, f, g, h] = det e3 e4
f1 f2 f3 f4
g1 g2 g3 g4
h1 h2 . h3 h4
(40)
Observing that a1 a2 a3 a4 = [aD, D, a 2 C, aC]/[aD, D, a 2 C, aC] = −[a 2 D, aD, D, aC]/[aD, a 2 C, aC, C]
(41)
we have |[aD, D, a 2 C, aC]|G given by [a 2 D, aD, aC, C][aD, D, a 2 C, aC] [aD, D, a 2 C, aC][aD, a 2 C, aC, C] .(42) [aD, a 2 C, aC, C][aD, D, a 2 C, aC] [aD, D, a 2 C, aC][aD, D, a 2 C, aC] Utilizing the identity relating the determinants derived in the Appendix, [aC, C, a 2 D, aD][a 2 C, aC, aD, D] = [C, aC, a 2 C, aD][C, aC, a 2 C, aD] + [C, aC, D, aD]2 , we have |[aD, D, a 2 C, aC]| G= [aD, D, aC, C]2
1 + γ γ¯ γ¯ δ γ δ¯ δ δ¯
(43)
,
(44)
where γ =
[aD, a 2 C, aC, C] , [aD, D, aC, C]
δ=
[aD, D, a 2 C, aC] . [aD, D, aC, C]
(45)
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S.A. Cherkis, N.J. Hitchin
Next we identify the differentials A0 and B0 in terms of the coordinates z(0), χ (0), dz(0) 1 dz(0) A0 = (2µ − l) , B0 = dχ (0) − L(0) − χ (0) (46) z(0) 2 z(0) From the definition of χ (17) we have k dQ Q Q z 1 dz dχ = 2 + − , − χ P P P z − qj 2 z
(47)
j =1
thus
k dQ Q Q z(0) dz(0) + − . B0 = 2 − L(0) P P P z(0) − qj z(0)
(48)
j =1
The conformal structure, from (44) can therefore be written as ¯ 0 )(γ A0 + δB0 ). ds 2 ∼ A0 A0 + (γ¯ A0 + δB
(49)
The volume corresponding to the expression on the right-hand-side is Vol = −
(2µ − l)2 |δ|2 ¯ dz ∧ d z¯ ∧ dQ ∧ d Q. z¯zP P¯
(50)
On the other hand the volume should be equal to 41 ω ∧ ω¯ for ω given by Eq. (9). Thus comparing the two expressions for ω = 2dQ ∧ dz/P we have the final metric z 2 (A0 A0 + (γ¯ A0 + δB ¯ 0 )(γ A0 + δB0 )). ds = (51) (2µ − l)δ 5. The Generalized Legendre Transform We have obtained the hyperk¨ahler metrics above by following the twistor space approach. In this section, we apply a different technique given by the generalized Legendre transform [14, 19]. It has been successfully applied in the case of D0 ALF in [13] to reproduce Atiyah-Hitchin metric, and to study Dk ALF metrics in [20]. We start with a brief outline of the construction. We use, as earlier, z as one of the complex coordinates on the fibres of the twistor space. In the patch ζ = ∞ it has the form z(ζ ) = z + vζ + wζ 2 − vζ ¯ 3 + z¯ ζ 4 .
(52)
The second coordinate χ (see (17)) is such that the holomorphic twistorial two-form has the form ω = dχ ∧ dz. If the section χ is represented by the function χ1 in the patch ζ = ∞ and χ2 in the patch ζ = 0, then we define a function fˆ and a contour C such that dζ ˆ dζ dζ f = χ − χ . (53) 1 j j −2 j 2 C ζ 0 ζ ∞ ζ
Gravitational Instantons of Type Dk
311
Given fˆ we define G such that ∂G/∂z(ζ ) = fˆ/ζ 2 . Next, we define the function of the coefficients of z(ζ ), 1 dζ F (z, v, w, v, ¯ z¯ ) = G. (54) 2πi C ζ 2 The generalized Legendre transform construction generates a formula for a K¨ahler potential K(z, z¯ , u, u) ¯ of the coordinates z and u. We impose the differential constraint ∂F /∂w = 0 which determines w as a function of z and v, and perform a Legendre transform on F with respect to coordinates v and v: ¯ K(z, z¯ , u, u) ¯ = F − vu − v¯ u, ¯ u=
∂F ∂F , u¯ = . ∂v ∂ v¯
(55)
In our case the above procedure produces C
dζ ˆ f = −4µ ζj
0
dζ ζ+ ζj
− 0
∞
dζ ζ j −2
(56)
χ1 ,
since we chose uij so that (12) holds, we have dζ z(ζ ) 1 2πi 0 ζ ζ 2 √ dζ z(ζ ) + (−1)l qi (ζ ) log( z(ζ ) + (−1)l qi (ζ )). ζ ζ
F (z, v, w, v, ¯ z¯ ) = −4µ −
k 1 2π i Cil i=1 l=0,1
A careful derivation of this formula can be found in [15]. For each i the pair (Ci0 , Ci1 ) consists of 4 figure-eight shaped contours on the Riemann surface η2 = z(ζ ), each contour surrounding two out of the eight points (η, ζ ) = (±qi (αij ), αij ). The projection of Ci0 on the ζ plane coincides with the projection of Ci1 . The two contours Ci1 have figure-eight shapes encircling points αij clockwise for odd j and counterclockwise for even j. With this in mind we obtain l=0,1
1 2π i
Cil
4 √ dζ (−1)j √ f (ζ ) log( z + (−1)l qi ) = 2 z j =1
For convenience let us define matrices Fvv Fvw Fv v¯ F = Fwv Fww Fwv¯ , Fvv Fv¯ v¯ ¯ Fvw ¯
αij
dζ √ f (ζ ). z
(57)
(58)
where the subscripts of F denote partial derivatives (e.g. Fvw = ∂ 2 F /∂v∂w), and vv vw v v¯ G G G G = Gwv Gww Gwv¯ = F −1 , (59) ¯ Gvw ¯ Gvv Gv¯ v¯
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where the superscripts merely label the components of G. The components of the metric are Kz¯z = Fz¯z − Fza Gab Fb¯z ,
(60) v v¯
Ku¯z = Fz¯ a G , Kuu¯ = −G , av
(61)
where a and b are summed over the values (v, w, v). ¯ And the metric is −1 ds 2 = v v¯ −Gv v¯ (Fz¯z − Fza Ga b Fb z¯ + Fza Ga v¯ (Gv v¯ )−1 Gvb Fb z¯ )dzd z¯ + G + (Fzb Gbv¯ dz − Gv v¯ du)(Fz¯ a Gav d z¯ − Gv v¯ d u) ¯ . (62) Since for a = v as well as for b = v¯ we have Gv v¯ Gab = Ga v¯ Gvb , we introduce also an index a taking values w, v¯ and b with values v, w. The unwieldy coefficient of dzd z¯ simplifies due to the identity (see the Appendix for the proof) Fz¯z − Fza Ga b Fb z¯ Fza Ga v¯ = 1, (63) det K(z,u) = det Gvb Fb z¯ −Gv v¯ and the metric has the following simple form ds 2 =
1 ¯ u) dzd z¯ + (αdz + βdu)(αd ¯ z¯ + βd ¯ , β
(64)
where α = Fzb Gbv¯ , β = −Gv v¯ .
(65)
To find expressions for the exact form of α, β, and u note that k √ dζ 1 1 u = Fv = − √ log( z − qi ), 1 2πi ζ z i=1 Ci
(66)
u¯ = Fv¯ =
(67)
k √ dζ ζ 2 1 √ log( z − qi ), 1 2πi ζ z i=1 Ci
Fw = −4µ −
k √ dζ 1 √ log( z − qi ). 1 2πi z i=1 Ci
(68)
If we introduce pn defined by pn =
√ k log( z − qi ) dζ ζ n+2 1 1 − +√ √ 1 2 2π i z z z − qi i=1 Ci
then
−p−2 −p−1 p0 = F = −p−1 −p0 p1 , p0 p1 −p2
(69)
G −1
(70)
Gravitational Instantons of Type Dk
313
and Fzv = −p−3 , Fzw = −p−2 , Fzv¯ = p−1 . Let D = det F, then 1 α= D
p−3 p−2 p−1 p−2 p−1 p0 , β = 1 D p−1 p0 p1
p−1 p0 p 0 p1 .
(71)
(72)
6. Comparison of the Two Approaches From the form of the one-forms A0 and B0 of Eq. (46) combined with the expression (15, 17) for χ we can identify u = χ (0) and A0 = (2µ − l)
dz dz , B0 = du − (L(0) − u) . z z
The two expressions (51),(64) for the metric coincide if δ eiφ , α = γ − (L(0) − u) 2µ − l δ β=z eiφ 2µ − l
(73)
(74) (75)
for some real-valued function φ. To start, we use the expressions (45) for δ and γ to find δ [1, a, aC, a 2 C] = , (76) 2µ − l [(2µ − l)a + a 2 C, 1, aC, C] (L(0) + u)δ [L(0) + u − C, a, aC, a 2 C] γ− =− . (77) 2µ − l [(2µ − l)a + a 2 C, 1, aC, C] On the other hand, introducing l = j =l (al − aj ) and using (69), one finds 1 al4 2µ − l p2 = − − Cl , z¯ z¯ l
(78)
1 aln+2 pn = − Cl , −2 ≤ n ≤ 1, z¯ l
(79)
4
l=1
4
l=1
1 1 1 1 1 =− (L(0) − χ (0)) − Cl , z¯ a1 a2 a3 a4 2 z¯ al l 4
p−3
(80)
l=1
which leads to the following expressions for the determinants (see the Appendix) p−3 p−2 p−1 [L(0) − 21 χ (0) − C, a, aC, a 2 C] 1 z¯ 3 p−2 p−1 p0 = , (81) [1, a, a 2 , a 3 ] p p p a1 a2 a3 a4 −1
0
1
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S.A. Cherkis, N.J. Hitchin
p−1 p1 − p02 =
z¯ 3 D = −
−1 [1, a, aC, a 2 C] , z¯ 2 [1, a, a 2 , a 3 ]
(82)
[(2µ − l)a + a 2 C, 1, aC, C] . [1, a, a 2 , a 3 ]
(83)
Indeed, recalling that a1 a2 a3 a4 = z/¯z we find z¯ [L(0) − 21 χ (0) − C, a, aC, a 2 C] , z [(2µ − l)a + a 2 C, 1, aC, C] [1, a, aC, a 2 C] β = z¯ . [(2µ − l)a + a 2 C, 1, aC, C] α=−
(84) (85)
Thus the relations (74) indeed hold with eiφ = z¯ /z. Acknowledgement. SCh was supported in part by DOE grant DE-FG02-90ER40542. NJH thanks the University of Bonn and SFB40 (now the Max-Planck-Institut) for its support 25 years ago.
Appendix Resultants. Here we find the condition on coefficients Ci and Di for the following system of equations to have a solution in ζ 4 j =1
Cj = 0, ζ − aj
4 j =1
Dj = 0. ζ − aj
To compare with the expressions of Sect. 4 put Cj = Cj daj and Dj = Dj daj . For convenience let us introduce polynomials h(ζ ) = 4i=1 (ζ − ai ) and gj (ζ ) = h(ζ )/(ζ − aj ). Then for M(ζ ) =
4
Ci gi (ζ ), N (ζ ) =
i=1
4
Di gi (ζ ),
(86)
i=1
the above system is equivalent to the system of two third order equations M(ζ ) = 0 = N(ζ ). M and N have no common root if and only if the six polynomials M(ζ ), ζ M(ζ ), ζ 2 M(ζ ), N (ζ ), ζ N (ζ ), ζ 2 N (ζ ) are linearly independent. These are fifth order polynomials and can be expanded in the basis formed by e.g. h(ζ ), ζ h(ζ ), gi (ζ ). It is convenient to introduce A0 =
4 i=1
Ci , B0 =
4
Di , A1 =
i=1
then ζ M(ζ ) = A0 h(ζ ) +
4 i=1
ai Ci , B1 =
4
ai D i ,
aj Cj gj (ζ ), ζ 2 M(ζ ) = A0 ζ h(ζ ) + A1 h(ζ ) + aj2 Cj gj (ζ ), with analogous expressions for N (ζ ).
(87)
i=1
(88) (89)
Gravitational Instantons of Type Dk
315
Now the condition for the existence of a solution is the vanishing of the Sylvester determinant B0 0 0 R= A0 0 0
B1 B0 0 A1 A0 0
a12 D1 a1 D 1 D1 a12 C1 a1 C1 C1
a22 D2 a2 D 2 D2 a22 C2 a2 C2 C2
a32 D3 a3 D3 D3 a32 C3 a3 C3 C3
a42 D4 a4 D 4 D4 . 2 a4 C4 a4 C4 C4
(90)
One finds R to be given by R = ij kl Ci Dj ak (A0 Dk − B0 Ck ) al2 (A0 Dl − B0 Cl ) − al (B1 Cl − A1 Dl ) . (91) Introducing A2 =
4
i=1 Ci /ai
and B2 =
4
B0 0 a12 D1 0 0 a1 D1 0 −B2 D1 R= A0 0 a12 C1 0 0 a C 1 1 0 −A C 2 1
i=1 Di /ai
a22 D2 a2 D 2 D2 a22 C2 a2 C2 C2
we have
a32 D3 a3 D 3 D3 a32 C3 a3 C3 C3
a42 D4 a4 D 4 D4 . a42 C4 a4 C4 C4
(92)
Determinant relations. Computing the determinant of the 8 × 8 matrix aC C 2 a D aD 0 D 0 0
0 aC C/a 0 aD 0 0 aD = D −aD D/a 0 C −aC aC −a 2 C
0 aC C/a 0 aD 0 0 aD D = 0 D/a 0 C 0 aC −a 2 C
0 C/a aD 0 D = 0. D/a C aC
(93)
On the other hand the same determinant equals −[aC, C, a 2 D, D][aC, C, D, D/a] + [C/a, C, aC, D][aC, a 2 D, aD, D] +[aC, C, aD, D]2 . (94) Thus [aC, C, a 2 D, aD][a 2 C, aC, aD, D] = [C, aC, a 2 C, aD][C, aC, a 2 C, aD] +[C, aC, D, aD]2 , (95) which produces Eq. (43) used in Sect. 4.
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The determinant of K(z,u) . The function z(ζ ) trivially satisfies the relations z(ζ )z¯z = −z(ζ )v v¯ , z(ζ )za = z(ζ )v(a −1) , z(ζ )b z¯ = −z(ζ )(b +1)v¯ , which implies analogous relations for F , det K(z,u) = −Gv v¯ Fz¯z + Fza Gv v¯ Ga b − Ga v¯ Gvb Fb z¯ . (96) For an n × n matrix A we denote by A(k) the k th compound matrix, which is the matrix composed of all order k minors of A. The adjugate compound matrix adj(k) is obtained from A(k) by replacing each k-minor by its complementary minor with the corresponding factor of (−1)l and transposition. In other words, due to the Laplace expansion of determinants, A(k) adj(k) A = det A 1. Thus from the Binet-Cauchy theorem −1 1 G(2) = F (2) = (97) adj(2) F. det F Thus utilizing the relations Fz¯z = −Fv v¯ , Fza = Fv(a −1) , and Fb z¯ = −F(b +1)v¯ we have 1 (1) det K(z,u) = (98) Fv v¯ Fv v¯ − Fv(a −1) F(b +1)v¯ adj(2) F(v,a ;b ,v) ¯ . det F The expression in brackets above is exactly the Cauchy expansion for det F. Thus we have det K(z,u) = 1. Relations between the pn and Cl expressions. We observe that [1, a, a 2 , a 3 ] = ij kl i j
ak − a j , aj − a i
(99)
without summation over the repeated indices. Then al − a i (2¯z)2 (p−1 p1 − p02 ) = ai Ci al2 Cl = i l i,l
=
−1 1 [1, a, aC, a 2 C] ij kl 2 . (a − a )a C a C = − k j i i l l 2 [1, a, a 2 , a 3 ] [1, a, a 2 , a 3 ]
(100)
Next note the identities
4 1, k = 3 k al = 0, 0 ≤ k ≤ 2 . l −1/a a a a , k = −1 l=1 1 2 3 4
Using these we have 1/a p−3 p−2 p−1 1 1 1 1 1 −(2¯z)3 p−2 p−1 p0 = det a diag( , , , )(C − L(0) 1 2 3 4 p−1 p0 p1 2 a 1 + χ (0), aC, a 2 C, a) 2 [L(0) − 21 χ (0) − C, a, aC, a 2 C] 1 =− , a1 a2 a3 a4 [1, a, a 2 , a 3 ]
(101)
Gravitational Instantons of Type Dk
317
as well as 1 1 1 1 1 a , , , ) C, aC, a 2 C + a(2µ − l), 1 (2¯z)3 D = det 2 diag( a 1 2 3 4 a3
=−
[(2µ − l)a + a 2 C, 1, aC, C] . [1, a, a 2 , a 3 ]
(102)
References 1. Hawking, S.W.: Gravitational Instantons. Phys. Lett. A 60, no.2, 81–83 (1977) 2. Atiyah, M., Hitchin, N.J.: The geometry and dynamics of magnetic monopoles. M. B. Porter Lectures. Princeton, NJ: Princeton University Press, 1988 3. Hitchin, N.J.: Twistor construction of Einstein metrics. Global Riemannian geometry (Durham, 1983), Ellis Horwood Ser. Math. Appl., Chichester: Horwood, 1984, pp. 115–125 4. Cherkis, S.A., Kapustin, A.: Singular monopoles and gravitational instantons. Commun. Math. Phys. 203, 713 (1999) 5. Gibbons, G.W., Hawking, S.W.: Action Integrals And Partition Functions In Quantum Gravity, Phys. Rev. D 15, 2752 (1977) 6. Hawking, S.W.: Euclidean Quantum Gravity. In: Recent Developments in Gravitation, Lectures presented at 1978 Cargese Summer School, Cargese, France, Jul 10-29, 1978, M. Levy, S. Deser, eds., New York: Plenum, 1979 7. Gibbons, G.W.: Gravitational Instantons: A Survey. Print-80-0056 (CAMBRIDGE) Review talk given at Int. Congress of Mathematical Physics, Lausanne, Switzerland, Aug 20-25, 1979 8. Sen, A.: A note on enhanced gauge symmetries in M- and string theory. JHEP 9709, 001 (1997) 9. Seiberg, N., Witten, E.: Gauge dynamics and compactification to three dimensions. In: The mathematical beauty of physics, (Saclay, France, 5-7 Jun 1996. (in memory of C. Itzykson)), J.B. Zuber, eds., Singapore: World Scientific, 1997, pp. 333–366 10. Cherkis, S.A., Kapustin, A.: Singular monopoles and supersymmetric gauge theories in three dimensions. Nucl. Phys. B 525, 215 (1998) 11. Hitchin, N.J.: Polygons and gravitons. Math. Proc. Cambridge Philos. Soc. 85, 465–476 (1979) 12. Abel, N.H.: Remarques sur quelques propri´et´es g´en´erales d’une certaine sorte de fonctions transendentales. J. Reine Angew. Math. Bd 3 (1828); Ouevres Compl`etes, Cristiania (1881) I, 444–456 13. Ivanov, I.T., Roˇcek, M.: Supersymmetric sigma models, twistors, and the Atiyah-Hitchin metric. Commun. Math. Phys. 182, 291 (1996) 14. Lindstr¨om, U., Roˇcek, M.: New Hyperk¨ahler Metrics And New Supermultiplets. Commun. Math. Phys. 115, 21 (1988) 15. Cherkis S.A., Kapustin, A.: Dk gravitational instantons and Nahm equations. Adv. Theor. Math. Phys. 2, 1287 (1999) 16. Hitchin, N.J.: Monopoles and Geodesics. Commun. Math. Phys. 83, 579 (1982) 17. Kronheimer, P.B.: Monopoles and Taub-NUT Metrics. M. Sc. Thesis, Oxford, 1985 18. Penrose, R.: Nonlinear gravitons and curved twistor theory. Gen. Rel. and Grav. 7, 31–52 (1976) 19. Hitchin, N.J., Karlhede, A., Lindstr¨om, U., Roˇcek, M.: Hyperk¨ahler Metrics And Supersymmetry. Commun. Math. Phys. 108, 535 (1987) 20. Chalmers, G., Roˇcek, M., Wiles, S.: Degeneration of ALF D(n) metrics. JHEP 9901, 009 (1999) Communicated by G.W. Gibbons
Commun. Math. Phys. 260, 319–392 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1406-6
Communications in
Mathematical Physics
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary Hans Lindblad Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA. E-mail: [email protected] Received: 8 September 2004 / Accepted: 14 March 2005 Published online: 2 September 2005 – © Springer-Verlag 2005
Abstract: We study the motion of a compressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler’s equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a “physical condition”, related to the fact that the pressure of a fluid has to be positive. 1. Introduction We consider Euler’s equations ρ ∂t + V k ∂k vj + ∂j p = 0, j = 1, . . . , n
in
D,
where
∂i = ∂/∂x i , (1.1)
describing the motion of a perfect compressible fluid body in vacuum: (∂t + V k ∂k )ρ + ρ div V = 0,
div V = ∂k V k
in
D,
(1.2)
where V k = δ ki vi = vk and we use the summation convention over repeated upper and lower indices. Here the velocity V = (V 1, . . . , V n ), the density ρ and the domain D = ∪0≤t≤T {t}× Dt , Dt ⊂ Rn are to be determined. The pressure p = p (ρ) is assumed to be a given strictly increasing smooth function of the density. The boundary ∂Dt moves with the velocity of the fluid particles at the boundary. The fluid body moves in vacuum so the pressure vanishes in the exterior and hence on the boundary. We therefore also require the boundary conditions on ∂D = ∪0≤t≤T {t}× ∂Dt : (∂t + V k ∂k )|∂ D ∈ T (∂D), p = 0, on ∂D.
The author was supported in part by the National Science Foundation.
(1.3) (1.4)
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Constant pressure on the boundary leads to energy conservation and it is needed for the linearized equations to be well posed. Since the pressure is assumed to be a strictly increasing function of the density we can alternatively think of the density as a function of the pressure and for physical reasons this function has to be non-negative. Therefore the density has to be a non-negative constant ρ 0 on the boundary and we will in fact assume that ρ 0 > 0, which is the case of liquid. We hence assume that p(ρ 0 ) = 0
and
p (ρ) > 0,
for
ρ ≥ ρ0,
where
ρ 0 > 0. (1.5)
From a physical point of view one can alternatively think of the pressure as a small positive constant on the boundary. By thinking of the density as a function of the pressure the incompressible case can be thought of as the special case of constant density function. The motion of the surface of the ocean is described by the above model. Free boundary problems for compressible fluids are also of fundamental importance in astrophysics since they describe stars. The model also describes the case of one fluid surrounded by and moving inside another fluid. For large massive bodies like stars gravity helps holding it together and for smaller bodies like water drops surface tension helps holding it together. Here we neglect the influence of gravity which will just contribute with a lower order term and we neglect surface tension which has a regularizing effect. Given a bounded domain D0 ⊂ Rn , that is homeomorphic to the unit ball, and initial data V0 and ρ0 , we want to find a set D ⊂ [0, T ] × Rn , a vector field V and a function ρ, solving (1.1)–(1.4) and satisfying the initial conditions {x; (0, x) ∈ D} = D0 , V = V0 , ρ = ρ0 on {0} × D0 .
(1.6) (1.7)
In order for the initial-boundary value problem (1.1)–(1.7) to be solvable initial data (1.7) has to satisfy certain compatibility conditions at the boundary. By (1.2),(1.4) also implies that div V ∂ D = 0. We must therefore have ρ0 ∂ D = ρ 0 and div V0 ∂ D = 0. 0 0 Furthermore, taking the divergence of (1.1) gives an equation for (∂t + V k ∂k ) div V in terms of only space derivatives of V and ρ, which leads to further compatibility conditions. In general we say that initial data satisfy the compatibility condition of order m if there is a formal power series solution in t, of (1.1)–(1.7) (ρ, ˜ V˜ ), satisfying (∂t + V˜ k ∂k )j (ρ˜ − ρ 0 ){0}×∂ D = 0, j = 0, . . . , m − 1. (1.8) 0
Let N be the exterior unit normal to the free surface ∂Dt . Christodoulou[C2] conjectured the initial value problem (1.1)–(1.8), is well posed in Sobolev spaces under the assumption ∇N p ≤ −c0 < 0,
on
∂D,
where
∇N = N i ∂x i .
(1.9)
Condition (1.9) is a natural physical condition. It says that the pressure and hence the density is larger in the interior than at the boundary. Since we have assumed that the pressure vanishes or is close to zero at the boundary this is therefore related to the fact that the pressure of a fluid has to be positive. In general it is possible to prove local existence for analytic data for the free interface between two fluids. However, this type of problem might be subject to instability in Sobolev norms, in particular Rayleigh-Taylor instability, which occurs when a heavier
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
321
fluid is on top of a lighter fluid. Condition (1.9) prevents Rayleigh-Taylor instability from occurring. Indeed, if this condition is violated Rayleigh-Taylor instability occurs in a linearized analysis. In the irrotational incompressible case the physical condition (1.9) always hold, see [W1, W2, CL], and [W1, W2] proved local existence in Sobolev spaces in that case. [W1, W2] studied the classical water wave problem describing the motion of the surface of the ocean and showed that the water wave is not unstable when it turns over. Ebin [E1] showed that the general incompressible problem is ill posed in Sobolev spaces when the pressure is negative in the interior and the physical condition is not satisfied. Ebin [E2] also announced a local existence result for the incompressible problem with surface tension on the boundary which has a regularizing effect so (1.9) is not needed then. In [CL], together with Christodoulou, we proved a priori bounds in Sobolev spaces in the general incompressible case of non-vanishing curl, assuming the physical condition (1.9) for the pressure. We also showed that the Sobolev norms remain bounded as long as the physical condition holds and the second fundamental form of the free surface and the first order derivatives of the velocity are bounded. Usually, existence follows from similar bounds for some iteration scheme, but the bounds in [CL] used all the symmetries of the equation and so only hold for modifications that preserve all the symmetries. In [L1] we showed existence for the linearized equations and in [L3] we proved local existence for the nonlinear incompressible problem with non-vanishing curl, assuming that (1.9) holds initially. For the corresponding compressible free boundary problem with non-vanishing density on the boundary, there are however in general no previous existence or well-posedness results. Relativistic versions of these problems have been studied in [C1, DN, F, FN, R] but solved only in special cases. The methods used for the irrotational incompressible case use that the components of the velocity are harmonic to reduce the equations to equations on the boundary and this does not work in the compressible case since the divergence is non-vanishing and the pressure satisfies a wave equation in the interior. To be able to deal with the compressible case one therefore needs to use interior estimates as in [CL, L1]. Let us also point out that in nature one expects fluids to be compressible, e.g.water satisfies (1.5), see [CF]. For the general relativistic equations there is no special case corresponding to the incompressible case. In [L2] we showed existence for the linearized equations in the compressible case and here we prove local existence for the nonlinear compressible problem: Theorem 1.1. Suppose that p = p(ρ) is a smooth function satisfying (1.5). Suppose also that initial data v0 , ρ0 and D0 are smooth satisfying the compatibility conditions (1.8) to all orders m and D0 is diffeomorphic to the unit ball. Then if the physical condition (1.9) hold at t = 0 there is a T > 0 such that (1.1)–(1.4) and (1.6)–(1.7) has a smooth solution for 0 ≤ t ≤ T . Furthermore (1.9) hold for 0 ≤ t ≤ T , with c0 replaced by c0 /2. A few remarks are in order. The existence of smooth solutions implies existence of solutions in Sobolev spaces if one has a priori bounds in Sobolev spaces. In the incompressible case we had already proven a priori energy bounds in Sobolev spaces in [CL] as well as a continuation result, that the solution remains smooth as long as the physical condition is satisfied and the solution is in C 2 . Similar bounds in Sobolev spaces hold also in the compressible case. We also remark that there are initial data satisfying the compatibility conditions to all orders, see Sect. 16. If only finitely many compatibility conditions are satisfied then we get existence in C k for some k. What then is essential
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is that the physical condition hold and this and the existence time only depends on a bound of finitely many derivatives of initial data. This makes it possible to construct a sequence of smooth solutions converging to a solution in Sobolev norms since we have a uniform lower bound for the existence times, in terms of the Sobolev norm. A few remarks about the proof are also in order. As in the incompressible case [L3] we will use the Nash-Moser technique to prove local existence. However, because of the presence of the boundary problem for the wave equation for the enthalpy one has to take as many time derivatives of the equations as space derivatives. Therefore one has to use interpolation in time as well and one might just as well do smoothing also in time in the application of the Nash-Moser technique although there is no loss of regularity in the time direction. Because of this all our constants will depend on a lower bound of the time interval. This will make it a bit more delicate since we will also need to choose a small time, in order that the physical and coordinate conditions should hold. However, at the same time certain estimates are more natural when one includes time derivatives up to the highest order. The plan of the paper is as follows. We will assume that the reader is somewhat familiar with the notation in [L3]. We will also assume the existence proofs given in [L1, L2, L3] for the inverse of the linearized operator so we will only prove improved estimates here. In Sect. 2 we formulate Euler’s equations in the Lagrangian coordinates and derive the linearized equations in these coordinates. Here we also define a modified linearized operator which is easier to first consider. In Sect. 3 we define the orthogonal projection onto divergence free vector fields, the normal operator and decompose the equation onto a divergence free part and a wave equation for the divergence. In Sect. 4 we construct the families of tangential vector fields, define the modified Lie derivatives with respect to these and calculate its commutators with the normal operator and other operators that occur in the linearized equation. In Sect. 5 we derive estimates of derivatives of a vector field in terms of the curl, the divergence and tangential derivatives or the normal operator. In Sect. 6 we give tame estimates for the Dirichlet problem, in Sect. 7 we give tame estimates for the wave equation and in Sect. 8 we give tame estimates for the divergence free part. Then in Sect. 9 we put these estimates together to get tame estimates for the inverse of the modified linearized operator. In Sect. 10 we give estimates of the enthalpy in terms of the coordinate. In Sect. 11 we show that the physical and coordinate conditions can be satisfied for small times if they hold initially. In Sect. 12 we then get tame estimates for the inverse of the linearized operator. In Sect. 13 we give tame estimates for the second variational derivative. In Sect. 14 we construct the smoothing operators needed for the Nash-Moser iteration. Finally, in Sect. 15 we construct the Nash-Moser iteration that proves local existence of a smooth solution. In Sect. 16 we show that one can construct a large class of initial data that satisfy the compatibility conditions to all orders. Most of the steps of the proof above works for the pressure of any smooth strictly increasing functions of the density. However, the estimates for the enthalpy in terms of the coordinate is simplified if one assume that the pressure is a linear function of the density and we first do the proof in this case and then in Sect. 17 give the additional estimates needed for the general case. 2. Lagrangian Coordinates and the Linearized Equation Let us introduce Lagrangian coordinates in which the boundary becomes fixed. Let be a unit ball in Rn and let f0 : → D0 be a diffeomorphism. By a theorem in [DM] the volume form κ0 = det (∂f0 /∂y) can be arbitrarily prescribed up to a multiplicative
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
323
constant and by a scaling of the equations we can also assume that the volume of D0 is that of the unit ball. Assume that v(t, x), p(t, x), (t, x) ∈ D are given satisfying the boundary conditions (1.3)–(1.4). The Lagrangian coordinates x = x(t, y) = ft (y) are given by solving dx/dt = V (t, x(t, y)),
y ∈ .
x(0, y) = f0 (y),
(2.1)
Then ft : → Dt is a diffeomorphism, and the boundary becomes fixed in the new y coordinates. Let us introduce the notation ∂ ∂ ∂ Dt = = + Vk k, (2.2) ∂t y=constant ∂t x=constant ∂x for the material derivative. The partial derivatives ∂i = ∂/∂x i can then be expressed in terms of partial derivatives ∂a = ∂/∂y a in the Lagrangian coordinates. We will use letters a, b, c, . . . , f to denote partial differentiation in the Lagrangian coordinates and i, j, k, . . . , to denote partial differentiation in the Eulerian frame. In these coordinates Euler’s equation (1.1) becomes ρDt2 xi + ∂i p = 0,
(t, y) ∈ [0, T ] × ,
(2.3)
(t, y) ∈ [0, T ] × .
(2.4)
and the continuity equation (1.2) become Dt ρ + ρ div V = 0,
Here the pressure p = p(ρ) is assumed to be a smooth strictly increasing function of the density ρ. With h, the enthalpy, i.e. h (ρ) = p (ρ)/ρ and h = 0 when p = 0, (2.3) becomes Dt2 xi + ∂i h = 0,
(t, y) ∈ [0, T ] × .
(2.5)
Since h is a strictly increasing function of ρ we can solve for ρ = ρ(h) as a function of h and with e(h) = ln ρ(h) (2.4) become Dt e(h) + div V = 0 .
(2.6)
Euler’s equations are now replaced by Dt2 xi + ∂i h = 0,
(t, y) ∈ [0, T ] × ,
∂i =
where
∂y a ∂ , ∂x i ∂y a
(2.7)
and taking the divergence of (2.7) using that [Dt , ∂i ] = −(∂i V k )∂k we obtain a wave equation for the enthalpy Dt2 e(h) − h − (∂i V k )∂k V i = 0, h = 0 . (2.8) ∂
Here e(h) is a given smooth strictly increasing function and h =
i
∂i2 h = κ −1 ∂a κg ab ∂b h ,
where
gab = δij
∂x i ∂x j , ∂y a ∂y b
(2.9)
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H. Lindblad
√ and g ab is the inverse of the metric gab and κ = det (∂x/∂y) = det g. The initial conditions are x t=0 = f0 , Dt x t=0 = v0 , (2.10) ht=0 = e−1 (ln ρ0 ), Dt ht=0 = − div V0 /e e−1 (ln ρ0 ) , (2.11) where e−1 is the inverse function of e(h). If (x, h) satisfies (2.7)–(2.8) with initial data of the form (2.10)–(2.11) then (x, h) also satisfies (2.6). By a theorem in [DM] the volume form can be arbitrarily prescribed so we can in fact choose it so κ0 = det (∂f0 /∂y) = 1/ρ0 , in which case e(h) = − ln κ, since this is true when t = 0 and since Dt ln κ = div V . Hence we are left we two independent initial data, f0 and v0 . In order for (2.8) to be solvable we must have the following condition on e(h) and the coordinate condition: a,b
c1−1 ≤ e (h) ≤ c1 , |g ab | + |gab | ≤ c12 ,
(2.12)
|∂x/∂y|2 + |∂y/∂x|2 ≤ c12
(2.13)
for some constant 0 < c1 < ∞. In order for (2.7) to be solvable we must have the physical condition: ∇N h ≤ −c0 < 0,
on
∂,
where
∇N = N i ∂x i .
(2.14)
We assume that e(h) is a given smooth function of h that satisfies (2.12). The condition on the metric is true initially since is diffeomorphic to some set in Rn but we have to assume that the condition (2.14) is true initially. By continuity, the conditions on the inverse of the metric and on the enthalpy are then true also for small times, with c1 replaced by 2c1 and c0 replaced by c0 /2. In the iteration we construct we have to make sure that the iterates are small enough and the time is small enough that these conditions remain true for all the iterates. This will be discussed in Sect. 11. In order to solve the wave equation on a bounded domain one also needs compatibility conditions on initial data. These conditions are so that the initial conditions are compatible with the boundary condition h∂ = 0. These conditions are: Dtk h∂ = 0, when t = 0, for k = 0, . . . , m − 1 . (2.15) For k = 0, 1 these are simply conditions on initial data and for k ≥ 2 one can use (2.8) and (2.7) to express it in terms of lower time derivatives of h, so it can be calculated in terms of the initial conditions. Equation (2.15) is called the mth order compatibility condition. We will assume that our initial data are smooth and satisfy the mth order condition for all m. This will be used to construct an approximate solution that satisfies the equation to all orders as t → 0, and in fact the initial conditions will be turned into an inhomogeneous term that vanishes to all orders as t → 0. This will also be discussed in the following sections. We prove the following theorem: Theorem 2.1. Suppose that initial data (v0 , f0 , ρ0 ) in (2.10)–(2.11) are smooth and the compatibility conditions (2.15) hold for all orders m. Suppose also that (2.13) and (2.14) hold when t = 0. Then there is T > 0 such that (2.7)–(2.8) have a smooth solution x ∈ C ∞ ([0, T ] × ), if e(h) is a smooth strictly increasing function with e(0) = 0. Furthermore (2.13) and (2.14) hold for 0 ≤ t ≤ T with c0 replaced by c0 /2 and c1 replaced by 2c1 .
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
325
We remark that if x ∈ C ∞ ([0, T ] × ) and the conditions in the theorem hold then (2.8) has a solution h ∈ C ∞ ([0, T ] × ). Theorem 1.1 follows from Theorem 2.1 since initially we assumed that D0 is diffeomorphic to the unit ball, so (2.13) holds initially. We will first prove Theorem 2.1 when e(h) = ch is a linear function of h. This corresponds to p = p(ρ) = c0 (ρ − ρ0 ), where c, ρ0 > 0 are positive constants. The result is true in general when the pressure is any strictly increasing smooth function of the density. Only the estimates for the enthalpy in terms of the coordinates have to be modified and we show how to do this in Sect. 17. The reason we first pick a linear function is that in this case we get a linear wave equation for the enthalpy and that simplifies the estimates and makes it more similar to the incompressible case where we have a linear elliptic equation for the pressure. Let us now define the Euler map that will be used to find the solution of Euler’s equations. A solution of Euler’s equations is given by (x, h) = 0, where = (0 , . . . , n ) is given by i (x, h) = Dt2 xi + ∂i h,
i = 1, . . . , n,
and 0 (x, h) = Dt2 e(h) − h − (∂i V k )∂k V i ,
h∂ = 0.
(2.16)
(2.17)
We have assumed that our initial conditions satisfy compatibility to all orders, i.e. there are smooth functions (x0 , h0 ) satisfying the initial conditions (2.10)–(2.11) and for all k ≥ 0, h0 ∂ = 0. (2.18) Dtk (x0 , h0 )t=0 = 0, In the process of solving (x, h) = 0 we will only consider functions (x, h) which have the same time derivatives as (x0 , h0 ) when t = 0 and which satisfy h∂ = 0. Let us therefore introduce the notation ∞ ([0, T ] × ) = {u ∈ C ∞ ([0, T ] × ); Dtk ut=0 = 0, for all k ≥ 0} , (2.19) C00 ∞ = C ∞ ([0, T ]×) and C ∞ = C ∞ ([0, T ]×). Equation (2.18) then and for short, let C00 00 ∞ and if x −x ∈ C ∞ then it follows that also (x, h ) ∈ C ∞ . says that (x0 , h0 ) ∈ C00 0 0 0 00 00 Then by [H1] we can solve the equation 0 (x, h) = 0, with initial conditions (2.11) and boundary conditions h∂ = 0 (The result in [H1] is formulated for vanishing initial conditions and instead an inhomogeneous term that vanishes to all orders as t → 0. However, ˜ = 0 (x, h˜ + h0 ) − 0 (x, h0 ) = ˜ 0 (h) we can turn the problem into this by considering ˜ This gives a solution in h˜ ∈ C ∞ satisfying −0 (x, h0 ), with vanishing initial data for h. 00 the boundary condition h˜ ∂ = 0). ∞ , we Let (x0 , h0 ) be the formal solution given above. For x satisfying x − x0 ∈ C00 now define the Euler map = (1 , . . . , n ) to be
i (x) = Dt2 xi + ∂i h,
i = 1, . . . , n ,
where h = (x) is given implicitly by solving Dt2 e(h) − h − (∂i V j )(∂j V i ) = 0,
h∂ = 0
(2.20)
(2.21)
with initial conditions (2.11). A solution of Euler’s equations is given by (x) = 0 .
(2.22)
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By the preceding argument we can, in fact, find a solution h to (2.21) such that h − h0 ∈ ∞ if x−x ∈ C ∞ . The reason we choose to consider the map (x) instead of (x, h) is C00 0 00 that we must make sure that h∂ = 0 and that the physical condition is satisfied, since the linearized operator is not invertible otherwise. Alternatively, one could also have tried to only consider h satisfying these conditions, but it seems much more difficult to preserve these conditions in the smoothing process used in the Nash-Moser iteration. The main work will now be to prove tame estimates for the inverse of the linearized operator. In the Nash-Moser iteration we will in fact consider ˜ (u) = (u + x0 ) − (x0 ) .
(2.23)
Let F = (x0 ), when t ≥ 0 and F = 0 when t < 0, and for δ ≥ 0 let Fδ (t, y) = ∞ . We will solve for u in: F (t − δ, y). Then Fδ ∈ C00 ˜ (u) = Fδ − F 0 .
(2.24)
The Nash-Moser theorem says that if the linearized operator is invertible and we have tame estimates for its inverse and for the second variation of the operator, then in fact ∞ . But the right-hand we have a solution of (2.24) if the right hand side is small in C00 ∞ side of (2.24) tends to zero in C when δ → 0, so (2.24) has a solution for some δ > 0 and hence (u + x0 ) = 0,
0 ≤ t ≤ δ.
(2.25)
As pointed out above at each step of the iteration we will only have functions u that vanish to all orders as t → 0. This condition can in fact be preserved by smoothing operators. In order to solve (x) = 0 we must show that the linearized operator is invertible. Let us therefore calculate the linearized equations. Let δ be a variation in the Lagrangian coordinates, i.e. derivative w.r.t. a parameter when t and y are fixed. Since [δ, ∂/∂y a ] = 0 it follows that ∂y a ∂ [δ, ∂i ] = δ i = −(∂i δx l )∂l , (2.26) ∂x ∂y a where we used the formula for the derivative of the inverse of a matrix δA−1 = −A−1 (δA)A−1 . It follows that [δ− δx k ∂k , ∂i ] = 0 and applying δ − δx k ∂k to (2.20) gives: Dt2 δxi − (∂k Dt2 xi )δx k − ∂i δx k ∂k h − δh = (x)δxi − δx k ∂k i . (2.27) It follows from this and applying δ to (2.21) that we have: Lemma 2.2. Let x = x(r, t,y) be a smooth function of (r, t, y) ∈ K = [−ε, ε]× [0, T ] × , ε > 0, such function of (r, t, y) ∈ K, that x r=0 = x. Then (x) is a smooth such that ∂(x)/∂r r=0 = (x)δx, where δx = ∂x/∂r r=0 and the linear map L0 = (x) is given by (x)δxi = Dt2 δxi + (∂i ∂k h)δx k − ∂i δx k ∂k h − δh , (2.28) where δh is a solution of
Dt2 (e (h)δh) − δx k ∂k Dt2 e(h) − δh − δx k ∂k h − 2(∂i V k )∂k (δV i − δx l ∂l V i ) = 0, δh = 0 . (2.29) ∂
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327
In order to use the Nash-Moser iteration scheme to obtain a solution of (2.22) we must show that the linearized operator is invertible and that the inverse satisfies tame estimates: Theorem 2.3. Suppose that x0 , h0 ∈ C ∞ [0, T ] × is a formal solution at t = 0, i.e. (2.18) hold. Suppose that (2.13) and (2.14) hold when t = 0. Let α |u |r,∞ = sup sup |∂t,y u(t, y)| . (2.30) 0≤t≤T y∈ |α|≤r
Then there is a T0 = T (x0 , h0 ) > 0, depending only on upper bounds for |x0 |r0 −1 +4,∞ + |h0 |r0 +4,∞ , where r0 = [n/2] + 1, c0 and c1 , such that the following hold. ∞ ∞ and If x − x0 ∈ C00 , h is the solution to (2.21) with h − h0 ∈ C00 T ≤ T0 ,
|x − x0 |r0 +4,∞ ≤ 1,
(2.31)
then (2.13) and (2.14) hold for 0 ≤ t ≤ T with c0 replaced by c0 /2 and c1 replaced by 2c1 . Furthermore, linearized equations (x)δx = δ,
in
[0, T ] × ,
(2.32)
∞ has a solution δx ∈ C ∞ . The solution satisfies the estimates where δ ∈ C00 00 a ≥ 0, |δx |a,∞ ≤ Ca |δ |a+r0 ,∞ + |δ |0,∞ |x − x0 |a+2r0 +4,∞ , (2.33)
where Ca = Ca (x0 , h0 , c0−1 , c1 ) is bounded when a is bounded. Furthermore is twice differentiable and the second derivative satisfies the estimates | (x)(δx, x) |a,∞ ≤ Ca |δx |a+2r0 +4,∞ | x |0,∞ + |δx |0,∞ | x |a+2r0 +4,∞ + |x − x0 |a+3r0 +6,∞ |δx |0,∞ | x |0,∞ .
(2.34)
Theorem 2.1 follows from Theorem 2.3 (or rather a version with H¨older norms) and the Nash-Moser theorem, Theorem 15.1. Theorem 2.3 follows from Lemma 11.3 and Theorem 12.2. More precisely, we will first prove an L2 estimate: δ x(t) ˙ r + δx(t) r ≤ Cr
r
|x |r+r0 +4−s,∞
s=1
where u(t) r =
t
δ(τ ) s dτ,
(2.35)
0
α ∂t,y u(t, ·) L2 () ,
(2.36)
|α|≤r
and we will first prove this estimate for a lower order modification, L1 , of the linearized operator to be described below. Furthermore in Theorem 9.1 we will first prove the estimate for L1 expressed in the Lagrangian coordinates, i.e. when also the vector field is expressed in the Lagrangian frame to be described below.
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H. Lindblad
It will follow from estimating the solution of the wave equation (2.29) that δh will have the same regularity as δx. Since [∂i , Dt2 ]δx i = 2(∂i V k )∂k (δV i − δx l ∂l V i ) + (∂k Dt V i )∂i δx k + δx l ∂l (∂i V k )∂k V i (2.37) we get by taking the divergence of (2.28): Dt2 div δx − δx l ∂l div Dt V − (∂i V k )∂k V i − δx k ∂k h − δh + 2(∂i V k )∂k ×(δV i − δx l ∂l V i ) = div ( (x)δx) − (∂i δx l )∂l i − δx l ∂l div . (2.38) Hence if we add (2.29) and (2.38) we get div ( (x)δx) = Dt2 div δx + e (h)δh + (∂i δx l )∂l i .
(2.39)
There is also a similar identity for the curl, see [L3]; curl ( (x)δx) = LDt curl Dt δx − δx k ∂vk + (∂i δx k )∂j k − (∂j δx k )∂i k , (2.40) where LDt is the space time Lie derivative with respect to the vector field Dt = (1, V ): LDt σij = Dt σij + (∂i V l )σlj + (∂j V l )σil
(2.41)
restricted to the space components. It can be integrated along characteristics since it is invariant under changes of coordinates: j j Dt aai ab σij = aai ab LDt σij , where aai = ∂x i /∂y a . (2.42) We now want to modify the linearized operator by adding a lower order term so as to remove the last term on the right in (2.39) or else replace it by a lower order term proportional to div δx and δh so that the equation div ( (x)δx) = 0 gives an estimate for div δx in terms of δh which we have better control of than ∂δx. In [L3] we used the modification L1 δx i = L0 δx i − δx l ∂l i + δx i div = Dt2 δxi − (∂k Dt2 xi )δx k − ∂i δx k ∂k h − δh + δx i div ,
(2.43)
div = Dt div V + h + (∂i V j )∂j V i = Dt div V + Dt2 e(h) .
(2.44)
where
Then we get div (L1 δx) = Dt2 div δx + e (h)δh + (div δx) div .
(2.45)
L1 is a lower order modification of the linearized operator, that reduces to the linearized operator at a solution of (x) = 0. We will first prove that L1 is invertible using that it has a nice equation for the divergence. Then, once we have inverted L1 we will, obtain estimates for L1 that are so good that they can be used to iterate and deal with lower order modifications like L0 . However, as pointed out in [Ha] one do not need invert the
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
329
operator exactly but only do so up to a quadratic error. But this operator L0 or some modification of it is likely to show up at other places so it seems important to show that a more general class of operators are invertible and to have good estimates for them. Things are somewhat easier to see if we express the vector field in the Lagrangian frame: Wa =
∂y a i δx , ∂x i
ωab =
∂x i ∂x i ∂ i v j − ∂ j vi . a b ∂y ∂y
(2.46)
Then ∂y a k δV − δx l ∂l V k , k ∂x k ∂y a 2 2 a k 2 l k Dt W = δx − δx ∂ D x − 2 δV − δx ∂ V v D . ∂ i k i l k i t t ∂x i Dt W a =
(2.47) (2.48)
Since also Dt gab =
∂x i ∂x i ∂ i vj + ∂ j vi , a b ∂y ∂y
(2.49)
the modified linearized operator (2.43) in the Lagrangian frame becomes L1 W a = gab Dt2 W b−∂a (∂c h)W c −δh + Dt gac − ωac Dt W c +div W a . (2.50) Also, the divergence is invariant, div δx = div W = κ −1 ∂a κW a .
(2.51)
Now, Dt does not commute with taking the divergence, if κ −1 Dt κ = div V = 0, so we will replace it by a modified time derivative that does: Dˆ t = Dt + div V , i.e. Dˆ t W a = Dt W a + (div V )W a = κ −1 Dt (κW a ). (2.52) It then follows that Dˆ t div W = div Dˆ t W
(2.53)
if σ = ln κ then σ˙ = Dt σ = div V , see [L1]. With this notation we have Dˆ t2 = (Dt + div V )(Dt + div V ) = Dt2 + 2σ˙ Dt + σ˙ 2 + σ¨ = Dt2 + 2σ˙ Dˆ t + σ¨ − σ˙ 2 , so Dt2 = Dˆ t2 − 2σ˙ Dˆ t + σ˙ 2 − σ¨ ,
Dt = Dˆ t − σ˙ .
Hence, with W˙ = Dˆ t W and W¨ = Dˆ t2 W , we can write Eq. (2.50) as L1 W a = W¨ a − g ab ∂b (∂c h)W c − δh − B1 W˙ a − B0 W a ,
(2.54)
(2.55)
where B1 W˙ a = −g ab (Dt gbc − ωbc )W˙ c + 2σ˙ W˙ a , B0 W a = g ab (Dt gbc − ωbc )σ˙ W c − (Dt2 e(h) + σ˙ 2 )W a . Taking the divergence of (2.55) gives div L1 W = Dˆ t2 div W − (∂c h)W c − δh − div B1 W˙ − div B0 W .
(2.56) (2.57)
(2.58)
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H. Lindblad
On the other hand by (2.29), Dt2 (e (h)δh) − δh − (∂c h)W c + div B1 W˙ − 2σ˙ div W˙ + div B0 W + (Dt2 e(h) + σ˙ 2 ) div W = 0 .
(2.59)
If we add (2.59) to (2.58) we get div L1 W = Dt2 div W + e (h)δh + div div W,
(2.60)
as it should be, by (2.45). With ϕ = div W + e (h)δh we can hence alternatively write (2.59): Dˆ t2 (e (h)δh) − δh − (∂c h)W c + div B1 W˙ + div B0 W − 2σ˙ ϕ˙ + (Dt2 e(h) + σ˙ 2 )ϕ − div e (h)δh = 0
(2.61)
or Dt2 (e (h)δh) − δh + ∂i (∂ i δx k )∂k h + (∂ i δx k )∂i ∂k h − 2(∂i V k )∂k δV i = 0, δh = 0 . (2.62) ∂
We now also want to express L0 = (x) is these coordinates. In order to do this we must transform the term δx k ∂k i in (2.25) to the Lagrangian frame. If a = i ∂y a /∂x i , then (δx k ∂k i )∂y a /∂x i = W c ∇c a , where ∇c is covariant differentiation, see e.g. [CL]. Hence by (2.43) L0 W a = L1 W a − B3 W a ,
where
B3 W a = −W c ∇c a + W a div . (2.63)
3. The Projection and the Normal Operator. The Energy and Curl Estimates Let us now also define the projection P onto divergence free vector fields by P U a = U a − g ab ∂b pU , pU = div U, pU ∂ = 0 .
(3.1)
(Here q = κ −1 ∂a κg ab ∂b q . ) P is the orthogonal projection in the inner product gab U a W b κdy (3.2) U, W =
and its operator norm is one: P W ≤ W .
(3.3)
For a function f that vanishes on the boundary define Af W a = g ab Af Wb , where Af Wa = −∂a (∂c f )W c − q ,
(∂c f )W c − q = 0, q = 0 . ∂
(3.4)
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331
This is defined for general vector fields but it is symmetric in the divergence free class. If U and W are divergence free then U, Af W = na U a (−∂c f )W c dS, (3.5) ∂
where n is the unit conormal. If f ∂ = 0 then −∂c f ∂ = (−∇N f )nc . It follows that Af is a symmetric operator on divergence free vector fields, and in particular A = Ah
(3.6)
is positive since we assumed that −∇N h ≥ c > 0 on the boundary. We have |U, Af W | ≤ ∇N f/∇N h L∞ (∂) U, AU 1/2 W, AW 1/2 .
(3.7)
Note also that Af only depends on ∇N f on the boundary so we can replace f by something with the same first order derivatives at the boundary that is supported in a neighborhood of the boundary. We now want to estimate the norm of Af . Now the projection has norm one so we can drop q in (3.4). If S is a tangential vector field, i.e. tangential to the boundary at the boundary, then S a ∂a (∂c f )W c ) = (∂c Sf )W c + (∂c f )LS W c , where the Lie derivative, LS is defined in the next section. Furthermore if R is the normal vector field then we can replace f by the distance d to the boundary times the value of ∇N f at the boundary extended to be constant along the normal. Replacing f by this function we see that ∂c f is then independent of the radial variable so R a ∂a (∂c f )W c ) = (∂c f )RW c . In conclusion we get that Af W ≤ C ∇N Sf L∞ (∂) W + C ∇N f L∞ (∂) ∂W , (3.8) S∈S
where S is a spanning set of tangential vector fields. Let us now also define the projected multiplication operators Mβ with a two form β by M β Wa = P (βab W b ) .
(3.9)
Since the projection has norm one it follows that Mβ W ≤ β ∞ W .
(3.10)
Furthermore we define the operator taking vector fields to one forms GWa = M g Wa = P (gab W b ) .
(3.11)
Then G acting on divergence free vector fields is just the identity I . Let L1 be the modified linearized operator in (2.55). We now want to project the equation L1 W = F
(3.12)
to the divergence free vector fields: We will decompose L1 into 3 parts. We write W = W0 + W1 ,
W0 = P W,
W1a = g ab ∂b q1 .
(3.13)
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H. Lindblad
Then if g˙ ab = Dˇ t gab , where Dˇ t = Dt − σ˙ , we have ∂a Dt q1 = Dt (gab W b ) = g˙ ab W1b + gab W˙ b and ∂a Dt2 q1 = g¨ ab W b + 2g˙ ab W˙ b + gab W¨ b . Hence 1
1
1
1
W¨ 1a = g ab ∂b Dt2 q1 − 2g ab g˙ bc W˙ 1c − g ab g¨ bc W1c .
(3.14)
Since the projection of a gradient of a function that vanishes on the boundary vanishes it follows that P W¨ 1a = B2 (W1 , W˙ 1 )a , where ab a c ˙ ˙ B2 (W1 , W1 ) = −P 2g g˙ bc W1 + g ab g¨ bc W1c .
(3.15)
Let L10 = P L1 . Since div W¨ 0 = 0 and δh vanishes the boundary it follows from projecting (2.55) that, L10 W0 = W¨ 0 + AW0 − B10 W˙ 0 − B00 W0 , L10 W1 = AW1 − B11 W˙ 1 − B01 W1 ,
(3.16) (3.17)
where Bi0 W = P Bi W,
B11 W a = P B1 W a + 2P g ab g˙ bc W c ), B01 W a = P B0 W a + P g ab g¨ bc W c .
(3.18)
Hence the projection of (3.12) becomes L10 W0 = −L10 W1 + P F.
(3.19)
Here, by (2.60) W1a = g ab ∂b q1 ,
q1 = ϕ − e (h)δh,
q1 ∂ = 0 ,
(3.20)
where Dt2 ϕ + div ϕ = div F + div e (h)δh,
(3.21)
and by (2.59) Dt2 (e (h)δh) − δh = (∂c h)W c − div B1 W˙ + 2σ˙ div W˙ − div B0 W − (Dt2 e(h) + σ˙ 2 ) div W .
(3.22)
Hence we have obtained a system of equations for (W0 , W1 , ϕ, δh). We now want to show the main idea of how to obtain estimates for the divergence free equation. Let W˙ = Dˆ t W = Dt W + (div V )W = κ −1 Dt (κW ) = Dt W + σ˙ W . Then W˙ is divergence free if W is divergence free. Let us now derive the basic energy estimate for equations of the form W¨ + AW = H,
(3.23)
where A is either the normal operator or the smoothed out normal operator. Now, for any symmetric operator B we have d ˙ , W, BW = 2W˙ , BW + W, BW dt
(3.24)
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333
where B˙ is the time derivative of the operator B considered as an operator from the divergence free vector fields to the one forms, see Sect. 4. B˙ is defined by (4.24) with T = Dt . For the two operators that we will consider here this is given by (4.39) and (4.40). Note that the projection in (4.24) comes up here since we take the inner product with a divergence free vector field in (3.24). Let the lowest order energy E0 = E(W ) be defined by E(W ) = W˙ , W˙ + W, (A + I )W .
(3.25)
Since W, W = W, GW , it follows that ˙ W˙ + W, (A˙ + G)W ˙ E˙ 0 = 2W˙ , W¨ + (A + I )W + W˙ , G .
(3.26)
It follows from (4.24)–(4.36) and (3.7) and (3.10) that ˙ | ≤ h/ ˙ h ∞ W, AW , |W, AW
˙ | ≤ g |W, GW ˙ ∞ W, W .
(3.27)
The last two terms are hence bounded by a constant times the energy so it follows that ˙ h ∞ + g |E˙ 0 | ≤ E0 2 H + c E0 , c = h/ ˙ ∞ + 2. (3.28) Now, let w = W , w˙ = W˙ and w¨ = W¨ , i.e. wa = gab W b , w˙ a = gab W˙ b and w¨ a = gab W¨ b . Observe that w˙ = Dt w, etc. since W˙ = Dˆ t W . In fact Dt wa = g˙ ab W b + w˙ a ,
Dt w˙ a = g˙ ab W˙ b + w¨ a ,
(3.29)
where g˙ ab = Dˇ t gab and W˙ a = Dˆ t W a . Now our equation says that w¨ + AW = H ,
H = −B0 W − B1 W˙ + F.
Since curl AW = 0 it follows that | curl w| ¨ ≤ C |∂W | + |W | + |∂ W˙ | + |W˙ | + | curl F | ,
(3.30)
(3.32)
and hence |Dt curl w| ≤ C | curl w| ˙ + |∂W | + |W | , |Dt curl w| ˙ ≤ C |∂W | + |W | + |∂ W˙ | + |W˙ | + | curl F | .
(3.32) (3.33)
4. The Tangential Vector Fields and Lie Derivatives Following [L1], we now construct the tangential vector fields, that are time independent expressed in the Lagrangian coordinates, i.e. that commute with Dt . This means that in the Lagrangian coordinates they are of the form S a (y)∂/∂y a . Furthermore, they will satisfy, ∂a S a = 0.
(4.1)
Since is the unit ball in Rn the vector fields can be explicitly given. The vector fields y a ∂/∂y b − y b ∂/∂y a
(4.2)
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H. Lindblad
corresponding to rotations, span the tangent space of the boundary and are divergence free in the interior. Furthermore they span the tangent space of the level sets of the distance function from the boundary in the Lagrangian coordinates d(y) = dist (y, ∂) = 1 − |y|
(4.3)
away from the origin y = 0. We will denote this set of vector fields by S0 . We also construct a set of divergence free vector fields that spans the full tangent space at distance d(y) ≥ d0 and that are compactly supported in the interior at a fixed distance d0 /2 from the boundary. The basic one is (4.4) h(y 3 , . . . , y n ) f (y 1 )g (y 2 )∂/∂y 1 − f (y 1 )g(y 2 )∂/∂y 2 , which satisfies (4.1). Furthermore we can choose f, g, h such that it is equal to ∂/∂y 1 when |y i | ≤ 1/4, for i = 1, . . . , n and so that it is 0 when |y i | ≥ 1/2 for some i. In fact let f and g be smooth functions such that f (s) = 1 when |s| ≤ 1/4 and f (s) = 0 when |s| ≥ 1/2 and g (s) = 1 when |s| ≤ 1/4 and g(s) = 0 when |s| ≥ 1/2. Finally let h(y 3 , . . . , y n ) = f (y 3 ) · · · f (y n ). By scaling, translation and rotation of these vector fields we can obviously construct a finite set of vector fields that span the tangent space when d ≥ d0 and are compactly supported in the set where d ≥ d0 /2. We will denote this set of vector fields by S1 . Let S = S0 ∪ S1 denote the family of tangential space vector fields and let T = S ∪ {Dt } denote the family of space time tangential vector fields. Let the radial vector field be R = y a ∂/∂y a .
(4.5)
∂a R a = n
(4.6)
Now
is not 0 but for our purposes it suffices that it is constant. Let R = S ∪ {R}. Note that R span the full tangent space of the space everywhere. Let U = S ∪ {R} ∪ {Dt } denote the family of all vector fields. Note also that the radial vector field commutes with the rotations [R, S] = 0,
S ∈ S0 .
(4.7)
Furthermore, the commutators of two vector fields in S0 is just ± another vector field in S0 . Therefore, for i = 0, 1, let Ri = Si ∪ {R}, Ti = Si ∪ {Dt } and Ui = Si ∪ {R} ∪ {Dt }. Let U = {Ui }M i=1 be some labeling of our family of vector fields. We will also use multindices I = (i1 , . . . , ir ) of length |I | = r. So U I = Ui1 · · · Uir and LIU = LUi1 · · · LUir , where LU is the Lie derivative, defined below. Sometimes we will write LIU , where U ∈ S0 or I ∈ S0 , meaning that Uik ∈ S0 for all of the indices in I . Let us now introduce the Lie derivative of the vector field W with respect to the vector field T ; LT W a = T W a − (∂c T a )W c .
(4.8)
We will only deal with Lie derivatives with respect to the vector fields T constructed in the previous section. For those vector fields T we have [Dt , T ],
and
[Dt , LT ] = 0.
(4.9)
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335
The Lie derivative of a one form is defined by LT αa = T αa + (∂a T c )αc .
(4.10)
The Lie derivative also commutes with exterior differentiation, [LT , d] = 0 so LT ∂a q = ∂a T q
(4.11)
if q is a function. The Lie derivative of a two form is given by LT βab = T βab + (∂a T c )βcb + (∂b T c )βac .
(4.12)
Furthermore if w is a one form and curl wab = dwab = ∂a wb − ∂b wa , then since the Lie derivative commutes with exterior differentiation: LT curl wab = curl LT wab .
(4.13)
We will also use that the Lie derivative satisfies the Leibniz rule, e.g. LT (αc W c ) = (LT αc )W c + αc LT W c ,
LT (βac W c ) = (LT βac )W c + βac LT W c . (4.14)
Furthermore, we will also treat Dt as if it was a Lie derivative and set LDt = Dt .
(4.15)
Now of course this is not a space Lie derivative. It can however be interpreted as a space time Lie derivative. But the important thing is that it satisfies all the properties of the other Lie derivatives we are considering. The reason we want to call it LDt is simply a matter of that we will apply products of Lie derivatives and Dt applied to the equation and since they behave in exactly the same way it is more efficient to have one notation for them. In [L1] we used extensively that the Lie derivatives with respect to the vector fields above preserve the divergence free condition. This is no longer true if κ is not a constant, since div U = κ −1 ∂a (κU a ). This is no longer the case if U is not divergence free. One could modify the vector fields by multiplying them by κ −1 . However, instead we will essentially multiply the vector field we apply them to with κ. The modified Lie derivative is now for any of our tangential vector fields defined by L˜ U W = LU W + (div U )W.
(4.16)
They preserve the divergence free condition, in fact div L˜ U W = U˜ div W,
where
U˜ f = Uf + (div U )f/
(4.17)
if f is a function. This definition is invariant and (4.17) holds for any vector field U . However, in general, since we are considering Lie derivatives only with respect to the vector fields constructed above and only expressed in the Lagrangian coordinates, the definition is simpler for us Lˆ U W = κ −1 LU (κW ) = LU W + (U σ )W,
where
σ = ln κ.
(4.18)
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H. Lindblad
Due to (4.1), div S = Sσ if S is any of the tangential vector fields and div R = Rσ + n, if R is the radial vector field. For any of the tangential vector fields it then follows that div Lˆ U W = Uˆ div W,
Uˆ f = Uf + (U σ )f.
where
(4.19)
This has several advantages. The commutators satisfy [Lˆ U , Lˆ T ] = Lˆ [U,T ] , since this is true for the usual Lie derivative. Furthermore, this definition is consistent with our previous definition of Dˆ t . However, when applied to one forms we want to use the regular definition of the Lie derivative. Also, when applied to two forms most of the time we use the regular definition: However, when applied to two forms it turns out to be sometimes convenient to use the opposite modification: Lˇ T βab = LT βab − (U σ )βab ,
Uˇ = U − (U σ ).
(4.20)
We will most of the time apply the Lie derivative to products of the form αa = βab W b : LT βab W b = (Lˇ T βab )W b + βab Lˆ T W,
(4.21)
since the usual Lie derivative satisfies the Leibniz rule. Using the modified Lie derivative we indicated in [L2] how to extend the existence theorem in [L1] to the case when κ is no longer constant, i.e. Dt σ = div V = 0. This will be carried out in more detail here. We will now calculate the commutator between Lie derivatives and the operators defined in the previous section, i.e. the normal operator and the multiplication operators. It is easier to calculate the commutator with Lie derivatives of these operators considered as operators with values in the one forms. The one form w corresponding to the vector fields W is given by lowering the indices wa = W a = gab W b .
(4.22)
For an operator B on vector fields we denote the corresponding operators with values in the one forms by B. These are related by BWa = gab BW b ,
BW a = g ab B a .
(4.23)
Most operators that we consider will map onto the divergence free vector fields so we will project the result afterwards to stay in this class. Furthermore, in order to preserve the divergence free condition we will use the modified Lie derivative. If the modified Lie derivative is applied to a divergence free vector field then the result is divergence free, so projecting after commuting does not change the result. As pointed out above, for our operators it is easier to commute Lie derivatives with the corresponding operators from the divergence free vector fields to the one forms. Let BT be defined by BT W a = P g ab LT BWb − B b Lˆ T W .
(4.24)
In particular if B is a projected multiplication operator B a W = P (βab W b ) = βab W b − ∂a q, where q vanishes on the boundary, is chosen so that div BW = 0 then LT B a W = βab Lˆ T W b + (Lˇ T βab )W b + ∂a T q,
(4.25)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
337
and if we project to the divergence free vector fields then the term ∂a T q vanishes since if T is a tangential vector field then T q = 0 as well. It therefore follows that BT is another projected multiplication operator: B T Wa = P (Lˇ T βab )W b . (4.26) In particular, we will denote the time derivative of an operator by B˙ = BDt and for a projected multiplication operator this is ˙ = BDt W = P ( Dˇ t βab )W b . BW (4.27) If B maps on to the divergence free vector fields Lˆ T BW a = Lˆ T (g ab B a W ) = (Lˆ T g ab )B a W + g ab LT B a W,
(4.28)
Here Lˆ T g ab = −g ac g bd Lˇ T gcd , and if B maps onto the divergence free vector fields then Lˆ T B is also divergence free, so the left hand side is unchanged if we do so and we get: Lˆ T BW a = −P g ab (Lˇ T gbc )BW c + P g ab LT B a W − B a Lˆ T W + B Lˆ T W a . (4.29) By (4.26) applied the Gab = P (gab W b ) we see that GT W = P (g ab Lˇ T gbc )W c ) so a the first term in the right of (4.29) is GT BW . The second term is by definition (4.24) BT W so we get Lˆ T BW = B Lˆ T W + BT W − GT BW.
(4.30)
The most important property of the projection is that it almost commutes with Lie derivatives with respect to tangential vector fields: I.e. let P ua = ua − ∂a pU . Then P LT P ua = P LT ua
(4.31)
since LT ∂a pU = ∂a T pU vanishes when we project again, since T pU vanishes on the boundary. We have just used this fact above. We have already calculated commutators between Lie derivatives and the projected multiplication operators, so let us now also calculate the commutator between the Lie derivative with respect to tangential vector fields and the normal operator. Recall that the normal operator is defined by Af W a = g ab Af Wb , where Af Wa = −∂a (∂c f )W c − q , (∂c f )W c − q = 0, q ∂ = 0, (4.32) and f is the function that vanished on the boundary. Hence since the Lie derivative commutes with exterior differentiation: LT Af Wa = −∂a (∂c f )Lˆ T W c + (∂c Tˇ f )W c + (∂c T σ )f W c − T q . (4.33) However, now since q vanishes on the boundary it follows that T q also vanishes on the boundary and so does (∂c T σ )f W c . Therefore the last two terms vanish when we project again so we get (4.34) P g ab LT Af Wb = P g ab Af Lˆ T Wb + P g ab ATˇ f Wb .
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H. Lindblad
Let us now change notation so A = Ah , where h is the enthalpy, see Sec. 3. Then we have just calculated AT defined by (4.24) to be AT = ATˇ h i, i.e. AT = ATˇ h ,
A = Ah .
if
(4.35)
In particular, if T = Dt is the time derivative we will use the notation A˙ = ADt which then is ˙ = ADt W = A ˇ W. AW Dt h
(4.36)
We can now also calculate higher order commutators: Definition 4.1. If T is a vector field let BT be defined by (4.24). If T and S are two tangential vector fields we define BT S = (BS )T to be the operator obtained by first using (4.24) to define BS and then define (BS )T to be the operator obtained from (4.24) with BS in place of B. Similarly if S I = S i2 · · · S ir is a product of r = |I | vector fields then we define (4.37) BI = · · · (BS i1 ) · · · S ik . If B is a multiplication operator BW a = P g ab βbc W c then (4.38) BI W = P g ab (Lˇ IT βbc )W c . In particular if G is the identity operator GW a = P g ab gbc W c then (4.39) GI W = P g ab (Lˇ IT gbc )W c . If A is the normal operator then
AI W a = P g ab ∂b (∂c Tˇ I p)W c .
(4.40)
With BT as in (4.4) we have proven that if B maps onto the divergence free vector fields then Lˆ T BW = BWT + BT W − GT BW, WT = Lˆ T W. (4.41) Repeating this gives for a product of modified Lie derivatives: Lˆ IT BW = cI1 ...Ik I GI3 · · · GIk BI1 WI2
WJ = Lˆ JT W,
(4.42)
where the sum is over all combinations of I = I1 + . . . + Ik , and cI1 ...Ik I are some constants, such that cI1 ...Ik I = 1 if I1 + I2 = I . Let us then also introduce the notation GI1 I2 I = cI1 ...Ik I GI3 · · · GIk ,
(4.43)
where the sum is over all combination such that I3 + . . . Ik = I − I1 − I2 . With this notation we can write (4.42), Lˆ IT BW = GI1 I2 I BI1 WI2 ,
(4.44)
where again GI1 I2 I = 1 if I1 + I2 = I . Also let ˜ I1 ...Ik I = 0, G
if
I2 = I,
and
˜ I1 ...Ik I = GI1 ...Ik I , G
otherwise . (4.45)
Then we also have ˜ I1 I2 I BI1 WI2 . Lˆ IT BW = BWI + G
(4.46)
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339
5. Estimates for Derivatives of a Vector Field in Terms of the Curl, the Divergence and Tangential Derivatives or the Normal Operator The first part of the lemma below says that one can get a pointwise estimate of any first order derivative of a vector field by the curl, the divergence and derivatives that are tangential at the boundary. The second part say that one can also get estimates in L2 with a normal derivative instead of tangential derivatives. The last part says that we can get the estimate for the normal derivative from the normal operator. The lemma is formulated in terms in the Eulerian frame, i.e. in terms the original Euclidean coordinates. Later we will reformulate it in the Lagrangian frame and then also get estimates for higher derivatives in a similar fashion. Lemma 5.1. Let N˜ be a vector field that is equal to the normal N at the boundary ∂Dt and satisfies |N˜ | ≤ 1 and |∂ N˜ | ≤ K. Let q ij = δ ij − N˜ i N˜ j . Then (5.1) |∂β|2 dx ≤ C q kl δ ij ∂k βi ∂l βj + | curl β|2 + | div β|2 , |∂β|2 dx ≤ C δ ij N˜ k N˜ l ∂i βk ∂j βl + | curl β|2 +| div β|2 +K 2 |β|2 dx. (5.2) Dt
Dt
Suppose that δ ij αj is another vector field that is normal at the boundary and let Aβi = ∂i (αk β k − q) and q be chosen so that div Aβ = 0 and q|∂ = 0. Then δ ij αk αl ∂i β k ∂j β l ≤ C δ ij Aβi Aβj + |α|2 | curl β|2 + | div β|2 + |∂α|2 |β|2 dx . Dt
Dt
(5.3) This lemma was proven in [L3]. We will now state some results that were proven in [L3] and give some further definitions. Definition 5.1. For V and V any of the family of vector fields introduced in Sect. 4, and for β a two form, a one form, a function or a vector field we define j |LIU β |, |β|V |LIU Dt β |, (5.4) |β|V r = r, s = |I |≤r, I ∈V
|I |≤r, I ∈V , j ≤s
where LIU is a product of Lie derivatives. Furthermore let |∂yα Dtk β|, |β|r,s = |β|r = |α|+k≤r
|∂yα Dtk β| ,
|α|≤r, k≤s
V
where |β| is the point wise norm. Furthermore let β 0 = 1, β 0 = 1 and
V V |β|V β s= |β|s1 · · · |β|sk . β s = s1 · · · |β|sk , s1 +...+sk ≤s,si ≥1
(5.5)
(5.6)
s1 +...+sk ≤s,si ≥1
We note that if β is a function then LU β = Uβ and in general it is equal to this plus terms proportional to β. Hence (5.4) is equivalent to |I |≤r, I ∈V |U I β| . In particular if R denotes the family of space vector fields then |β|R r is equivalent to |β|r with a constant of equivalence independent of the metric. Note also that if β is the one form I q|. = |∂U βa = ∂a q then LIU β = ∂U I q so |∂q|V r |I |≤r, I ∈V
340
H. Lindblad
Definition 5.2. Let c1 be a constant such that |∂x/∂y|2 + |∂y/∂x|2 ≤ c12 . |gab | + |g ab | ≤ c12 ,
(5.7)
a,b
Let Ki0 respectively Ki , for i ≥ 1, denote continuous increasing functions of c1 + |x |i,0,∞ ,
respectively
|x |i,∞ + c1 + T −1 ,
(5.8)
which also depends on the order r of differentiation. Here, the norms are as in Definition 5.3. We note that the bounds for the metric g and its inverse follows from the bounds for the Jacobian of the coordinate and its inverse since gab = δij (∂x i /∂y a )(∂x j /∂y b ) a i b j and g ab = δ ij (∂y √ /∂x )(∂y /∂x ). It also follows that we have a bound for κ = det (∂x/∂y) = det g and its inverse. Furthermore, if we have a bound for κ −1 the bound for the inverse of the metric follows from the bound for the metric, since an upper bound for the eigenvalues and a lower bound for their product gives a lower bound for the eigenvalues, since they are all positive by assumption. Moreover we note that also a lower bound |∂x/∂y|2 ≥ c1−2 follows. In what follows it will be convenient to consider the norms of Lˆ IU W = κ −1 LIU (κW ) if W is a vector field and of Lˇ IU g = κLIU (κ −1 g), if g is the metric. The reason for this is simply that div(Lˆ IU W ) = Uˆ I div W and LIU curl w = curl (LIU W ) and when we lower indices wa = gab W b = (κ −1 gab )(κW b ) and apply the Lie derivative to the product we get LU wa = (Lˇ U gab )W b + gab Lˆ U W b . The following lemma was proven in [L3] for the cases without time derivatives. The cases with time derivatives follow from these by applying them to W replaced by Dˆ tk W , noting that div Dˆ t W = Dˆ t div W and curl (Dˆ tk W ) is equal to Dtk curl w plus terms that are lower order in time. This is similar to the calculation in [L1, L3] that curl (Lˆ IS W ) , where (Lˆ IS W )b = gab Lˆ IS W b is equal to LIS curl w plus terms of lower order. We have: b Lemma 5.2. Let W be a vector √ field and let wa = gab W be the corresponding one form. Let κ = det (∂x/∂y) = det g. Then
|κ| + |κ −1 | ≤ K10 ,
|U I κ| + |U I κ −1 | ≤ K10 cI1 ...Ik |U I1 g|· · · |U Ik g| . (5.9)
where the sum is over all I1 + . . . + Ik = I and K10 is as in Definition 5.2. With notation as in Definition 5.1 and Sect. 4 we have r−1 R R S R |κW |R | curl w| ≤K + |κ div W | + |κW | + |g/κ|R 10 r r r−s |κW |s . (5.10) r−1 r−1 s=0
We also have |κW |R r
≤ K10
r
g/κ
R r−s
R S | curl w|R s−1 + |κ div W |s−1 + |κW |s ,
s=0
where the first two terms in the sum should be interpreted as 0 if s = 0.
(5.11)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
341
The inequalities (5.10)–(5.11) also hold with (R, S) replaced by (U, T ) and moreover, R R S |κW |R r, s ≤ K10 | curl w|r−1,s + |κ div W |r−1,s + |κW |r,s R + |g/κ|U r+s−i−j |κW |i,j
(5.12)
i+j ≤r+s−1,i≤r,j ≤s
and |κW |R r,s ≤ K10
g/κ
U
i≤r,j ≤s
r+s−i−j
R S | curl w|R i−1,j+ |κ div W |i−1,j + |κW |i,j , (5.13)
where the sums are only over positive indices, and if i = 0 then the first two terms should be interpreted as 0. Definition 5.3. For V any of the family of vector fields in Sect. 4 let
W V r =
W V r,∞ =
LIU W ,
|I |≤r, I ∈V
LIU W ∞ ,
(5.14)
Dtk ∂yα W ∞ ,
(5.15)
|I |≤r, I ∈V
and let
W r =
Dtk ∂yα W ,
W r,∞ =
|α|+k≤r
|α|+k≤r
where W = W (t) = W (t, ·) L2 () , W ∞ = W (t) ∞ = W (t, ·) L∞ () . Let the mixed norms be defined by W V r,s =
s
Dt W V r , j
W r,s =
s
j
Dt ∂yα W
(5.16)
j =0 |α|≤r
j =0
and W V r,s,∞
=
s
j Dt
W V r,∞ ,
W r,s,∞ =
s
j
Dt ∂yα W ∞ . (5.17)
j =0 |α|≤r
j =0
Furthermore, let |α |r,∞ = sup α(t, ·) r,∞ ,
|α |r,s,∞ = sup α(t, ·) r,s,∞ ,
0≤t≤T
(5.18)
0≤t≤T
and
β r,∞ =
r1 +...+rk ≤r, ri ≥1
|β |r1 ,∞ · · · |β |rk ,∞ .
(5.19)
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H. Lindblad
It follows from the discussion after Definition 5.1 and the beginning of Lemma 5.3 that W r is equivalent to W R r with a constant of equivalence depending only on the dimension. As with the pointwise estimates it will sometimes be convenient to instead use Lˆ IU W = κ −1 LIU (κW ) . This in particular is true for the family of space tangential vector fields S. However instead of introducing a special notation we then write κW S r ; since κ is bounded from above and below by a constant K1 this is equivalent with a constant of equivalence K1 . Furthermore, by interpolation S S S κW S r ≤ K1 ( |g |r W + W r ) and W r ≤ K1 ( |g |r W + κW r ), and in our inequalities we will have lower order terms of this form anyway. Note also that it follows from the interpolation inequalities in Lemma 5.7 that
g r,∞ ≤ K1 |g |r,∞ . (5.20) | g r |0,∞ ≤ Here the constant K1 depends on a lower bound for T > 0 and in fact tends to infinity as T → 0. Most of the time this does not matter, but when estimating lower norms of the enthalpy we can not use this estimate. The following lemmas were proven in [L3] for the cases without time derivatives. Their generalizations to including time derivatives are immediate. Lemma 5.3. We have with a constant K10 as in Definition 5.2, κW r,0 ≤ K10 curl w r−1,0 + κ div W r−1,0 + κW S r,0 +K10
r−1
g r−s,0,∞ κW s,0
(5.21)
s=0
and κW r,0 ≤ K10
r
g r−s,0,∞ curl w s−1,0
s=0
+ κ div W s−1,0 + κW S s,0 ,
(5.22)
where the two terms with s − 1 should be interpreted as 0 when s = 0. With a constant K1 as in Definition 5.2, we have κW r ≤ K10 curl w r−1 + κ div W r−1 + κW Tr +K10
r−1
g r−s,∞ κW s
(5.23)
s=0
and r
g r−s,∞ curl w s−1 + κ div W s−1 + κW Ts , (5.24) κW r ≤ K10 s=0
where the two terms with s − 1 should be interpreted as 0 when s = 0. Furthermore κW r,s ≤ K10 curl w r−1,s + κ div W r−1,s + κW S r,s
+ g r+s−i−j,∞ κW i,j (5.25) i≤r, j ≤s, i+j ≤r+s−1
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
and
343
g r+s−i−j,∞ curl w i−1,j κW r,s ≤ K10 i≤r, j ≤s
+ κ div W i−1,j + κW S i,j ,
(5.26)
where the two terms with i − 1 should be interpreted as 0 if i = 0. The following lemma, proven in [L3], gives a bound of derivatives of a vector field by the curl, the divergence and the normal operator. We have: Lemma 5.4. Let c0 > 0 be a constant such that |∇N h| ≥ c0 > 0, let K1 and K2 be constants such that ∇N h L∞ (∂) +c1 ≤ K1 and S∈S ∇N Sh L∞ (∂) +|∂g|+c1 +K1 ≤ K2 . Then c0 ∂W ≤ AW + K1 ( curl w + div W ) + K2 W . (5.27)
By Lemma 5.4 we also have c0 ∂ Lˆ JT W ≤K2 curl (Lˆ JT W ) + div Lˆ JT W + ALˆ JT W + Lˆ JT W . (5.28) Here (Lˆ JT W )b = gab Lˆ JT W b , and as in the proof of Lemma 5.3 in [L3] we see that curl (Lˆ JT W ) is equal to curl LJT w = LJT curl w, plus terms of lower order, where wa = gab W b . In particular we see that we can get any space tangential derivative in this way so we also get: Lemma 5.5. With K2 as in Lemma 5.4 we have c0 W r,0 ≤ K2 curl w r−1,0 + div W r−1,0 + W S r−1,A +
r−1
g ∞,r−s W s,0 ,
(5.29)
s=0
where W S s,A =
ALˆ IS W .
(5.30)
|I |=s,I ∈S
We also have
c0 W r ≤ K2 curl w r−1 + div W r−1 + W Tr−1,A + W˙ Tr−1 +
r−1
|g |∞,r−s W s ,
(5.31)
s=0
where W Ts,A =
ALˆ IT W .
(5.32)
|I |=s,I ∈T
Let us now state the interpolation inequalities that we will use. For a proof of Lemma 5.6, see e.g. [H1, H2] for the L∞ estimate and [CL] for the L2 estimate.
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H. Lindblad
Lemma 5.6. There are constantsCr , respectively Cr,T , depending on r, respectively on r and T , such that if β, W ∈ C ∞ [0, T ] × is a two form, a function or a vector field, then 1−s/r
s/r
|β |s,∞ ≤ Cr,T |β |0,∞ |β |r,∞ , W s ≤
1−s/r s/r Cr W 0 W r .
(5.33) (5.34)
∞ ([0, T ] × ), see (2.19), (5.33) holds Here W s = W (t) s . Furthermore, for β ∈ C00 with constants independent of T .
The consequence we will use is that Lemma 5.7. There are constants C depending on T and r, such that if f, fi , α, β, W ∈ C ∞ [0, T ] × are functions, two forms or vector fields, then |f |s1 ,∞ · · · |f |sk ,∞ ≤ C |f |k−1 (5.35) 0,∞ |f |r,∞ , |α |r−s,∞ |β |r,∞ ≤ C |α |0,∞ |β |s,∞ + |β |0,∞ |α |r,∞ , (5.36) |β |r−s,∞ W s ≤ C |β |0,∞ W r + |β |r,∞ W 0 , (5.37) |f1 |s1 ,∞ · · · |fk |sk ,∞ ≤ C
k
|f1 |0,∞ · · · |fi−1 |0,∞
i=1
×||fi |r,∞ |fi+1 |0,∞ · · · |fk |0,∞ ,
(5.38)
where r = s1 + . . . + sk . Here W s = W (t) s . Furthermore, for α, β, f, fi ∈ ∞ ([0, T ] × ), see (2.19), (5.35)–(5.38) holds with constants independent of T . C00 Proof. This follows from using Lemma 5.6 on each factor and then using the inequality As/r B 1−s/r ≤ A + B, see [L3]. 6. Tame Estimates for the Dirichlet Problem In this section we will give tame estimates for the Dirichlet problem q = f, in [0, T ] × , q ∂ = 0.
(6.1)
Let us recall the definition of the mixed norms Dtk ∂yα q . q r,s =
(6.2)
|α|≤r, k≤s
In the proofs that follow we will use the interpolations |g |s,∞ |g |r,∞ ≤ K1 |g |s+r,∞ , |g |s+1,∞ |g |r+1,∞ ≤ K2 |g |s+r+1,∞ , where K1 and K2 are as in Definition 5.2.
(6.3)
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345
Theorem 6.1. Suppose that q is a solution of the Dirichlet problem, q ∂ = 0 and W1a = g ab ∂b q, then
W1 r+1,s + ∂q r+1,s ≤ K10 g r+s−i−j,∞ q i,j i≤r, j ≤s
+K10
s
g r+s+1−j,∞ W1 0,j
(6.4)
j =0
if r, s ≥ 0, and
W1 r+1,s + q r+2,s ≤ K20
(∂g, Dt g)
i≤r, j ≤s
r+s−i−j
q i,j .
(6.5)
Moreover if P is the orthogonal projection onto divergence free vector fields and W is any vector field, then
P W r,s ≤ K10 (6.6) g r+s−i−j,∞ W i,j . i≤r, j ≤s
Here K10 and K20 are as in Definition 5.2,
g r,∞ = |g |r1 ,∞ · · · |g |rk ,∞ ≤ K1 |g |r,∞
(6.7)
r1 +...+rk ≤r, ri ≥1
and
(∂g, Dt g) r,∞ =
|g |r1 +1,∞ · · · |g |rk +1,∞
r1 +...+rk ≤r, ri ≥1
≤ K2 |g |r+1,∞ .
(6.8)
Remark. Note that the constant in (6.4)–(6.6) are independent of T whereas those in (6.7)–(6.8) depend on a lower bound for T > 0. First two useful lemmas:
Lemma 6.2. Suppose that S ∈ S and q ∂ = 0, and Lˆ S W a = g ab ∂b q + F a .
(6.9)
Lˆ S W ≤ K10 ( div W + F ).
(6.10)
Then
Proof. Let WS = Lˆ S W , gab WSa WSb κ dy = WSa ∂a q κdy + WSa gab F b κdy.
(6.11)
If we integrate by parts in the first integral on the right, using that q vanishes on the boundary we get WSa ∂a q κ dy = − div (WS ) q κ dy. (6.12)
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H. Lindblad
However div WS = Sˆ div W . Then we can in integrate by parts athe angular direction. ˆ ) κdy = S = S a ∂a , Sˆ = S + div S so we get (Sf ∂ a S f κ) dy = 0, where ∂a S a = 0. Hence we get a WS ∂a q κ dy = div (W ) (Sq)κ dy. (6.13)
Here |Sq| ≤ |∂q| ≤ K10 (|WS | + |F |) so it follows that WS 2 ≤ K10 div W WS + F + K10 WS F ,
(6.14)
and using that the inequality A2 ≤ K(A + B)B implies that A ≤ (2K + 1)B we get for some other constant K10 : WS ≤ K10 ( div W + F ),
(6.15)
Lemma 6.3. Let W a = g ab wb Then Dˆ tk W a = g ab Dtk wb −
k−1 k
i
i=0
k
Proof. We have Dtk wb = Dt κ −1 gbc κW c which proves the lemma.
g ab (Dˇ tk−i gbc )Dˆ ti W c .
=
(6.16)
k k−i −1 i c ˆ i=0 i Dt (κ gbc ) Dt (κW ),
k
Proof of Theorem 6.1 . To simplify notation let us denote W a = W1a = g ab ∂b q. If we apply LIT to wa = gab W b we get ∂a T I q = gab WIb + c˜I1 I2 gI1 ab WIb2 ,
WI = Lˆ IT W,
gI ab = Lˇ IT gab , (6.17)
and the sum is over all combinations I = I1 + I2 , c˜I1 I2 are constants such that c˜I1 I2 = 0 if I2 = I . We assume that Lˆ IT = Lˆ JS Lˆ sDt , where J ∈ S and |J | = r + 1. If we write Tˆ I = Sˆ Tˆ K , WI = Lˆ S WK , and use Lemma 6.2 we get, since div WK = Tˆ K div W = Tˆ K q = κ −1 T K (κ q) , WI ≤ K10 Tˆ K q + K10 c˜I1 I2 gI1 ∞ WI2 ,
(6.18)
or if we sum over all of them, κW S r+1,s ≤ K10 κ q r,s + K10
g r+s+1−i−j,∞ κW S i,j , (6.19)
i≤r+1, j ≤s, i+j ≤r+s
We now want to apply Lemma 5.3 to W a = g ab ∂b q. Then curl w = 0 and div W = q, κW r+1,s ≤ K10 κ q r,s + K1 W S r+1,s
g r+s+1−i−j,∞ κW i,j . +K10 i≤r+1, j ≤s, i+j ≤r+s
(6.20)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
347
Equations (6.19) and (6.20) therefore give for r, s ≥ 0, κW r+1,s ≤ K10 κ q r,s
+K10 g r+s+1−i−j,∞ κW i,j .
(6.21)
i≤r+1, j ≤s, i+j ≤r+s
Using induction it follows that
g r+s−i−j,∞ κ q i,j
κW r+1,s ≤ K10
i≤r, j ≤s s
+K10
j =0
g r+s+1−j,∞ κW 0,j .
(6.22)
Equation (6.4) follows from this and ∂a q = gab W b . (Replacing κ by 1 just causes more terms of the same form.) To prove (6.5) we need estimates for the last terms on the right in (6.4). By (6.4) with r = 0 we have s s
∂ 2 q 0,s ≤K10 g s−j,∞ q 0,j + K10 g s+1−j,∞ ∂q 0,j . (6.23) j =0
j =0
s
Since Dˆ ts q = κ −1 ∂a Dt κg ab ∂b q we have Dts q ≤ Dˆ ts q + K10
s−1
j g s−j,∞ ∂ 2 Dt q j =0
j + g s+1−j,∞ ∂Dt q ,
(6.24)
and using (6.23) it follows that Dts q
s
g s−j,∞ q 0,j ≤ K10 j =0
+K10
s−1
g s+1−j,∞ ∂q 0,j .
(6.25)
j =0
We have g ab (∂a q)(∂b q) κ dy = − ( q)q κ dy and there is a constant K10 , depending just on the volume of , i.e. κ dy, such that q ≤ K10 q , see [SY]. Hence in addition we have (6.26) if q ∂ = 0. q + ∂q ≤ K10 q , Equation (6.26) applied to Dts q in place of q together with (6.25) gives ∂q 0,s ≤ K10
s
g s−j,∞ q 0,j j =0
+K10
s−1
g s+1−j,∞ ∂q 0,j , j =0
(6.27)
348
H. Lindblad
where the last term should be interpreted as 0 if s = 0, and inductively it follows that ∂q 0,s ≤ K20
s
j =0
(∂g, Dt g) s−j,∞ q 0,j .
(6.28)
It remains to prove the estimates for the projection (6.6). We have W = W0 + W1 , where W0 = P W , and W1 = g ab ∂b q, where q = div W and q ∂ = 0. Proving (6.6) for r ≥ 1 reduces to proving it for r = 0 by using (6.4), since Rˆ I q = div (Lˆ IR W ) and replacing κ by 1 just produces more terms of the same form . Equation (6.6), for r = 0 and s = 0 follows since the projection has norm 1, P W ≤ W . Since the projection of g ab Dtk w1b = g ab ∂b Dtk q vanishes we obtain from Lemma 6.3: P Dˆ tk W1 ≤ K1
k−1
g k−i Dˆ ti W1 .
(6.29)
i=0
Since also P Dˆ tk W0 = Dˆ tk W0 we have (I − P )Dˆ tk W1 = (I − P )Dtk W,
(6.30)
and hence since the projection has norm one Dˆ tk W1 + Dˆ tk W0 ≤ K1 Dˆ tk W + K1
k−1
g k−i Dˆ ti W1 .
(6.31)
i=0
Using induction we therefore obtain Dˆ tk W0 + Dˆ tk W1 ≤ K1
k
j g k−j,∞ Dˆ t W .
(6.32)
j =0
Since as before replacing κ by 1 just produces more terms of the same form, this proves (6.6) also for r = 0. Equations (6.7)–(6.8) follow from interpolation. 7. Tame Estimates for the Wave Equation Existence of solutions for a wave equation with Dirichlet boundary conditions and initial conditions satisfying some compatibility conditions is well known, see e.g. [H3]. The result in [H3] is stated with vanishing initial condition but we will reduce to that case by subtracting a function that solves the equation to all orders as t → 0. We consider the Cauchy problem for the wave equation on a bounded domain with Dirichlet boundary conditions: Dt2 (e ψ) − ψ = f, in [0, T ] × , ψ ∂ = 0, (7.1) ψ = ψ0 , Dt ψ = ψ1 (7.2) t=0
t=0
Here ψ = √
1 ∂a det gg ab ∂b ψ . det g
(7.3)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
349
where g ab is the inverse of the metric gab and det g = det{gab } = κ 2 , in our earlier notation. We assume that e is positive, g ab is symmetric and positive definite (since the metric is), and that g ab and e are smooth satisfying: e + 1/e ≤ c1 , (|g ab | + |gab |) ≤ c12 , (7.4) a,b
for some constants 0 < c1 < c1 < ∞. We will apply this theorem to Dt2 (e (h)δh) − δh = −∂i (∂ i δx k )∂k h −(∂ i δx k )∂i ∂k h + 2(∂i V k )∂k δV i ,
δh = 0 ∂
(7.5)
with vanishing initial conditions and the right hand side vanishing to all orders as t → 0. We will also need some estimates for the equation for the enthalpy itself, which is best dealt with by writing h = h˜ + h0 , where h0 is the smooth approximate solution. In particular, in case e(h) = c−2 h we get the equation c−2 Dt2 h˜ − h˜ = − c−2 Dt2 h0 − h0 − (∂i V j )(∂j V i ) (7.6) with vanishing initial conditions but where the right hand side vanishes to all orders as t → 0 and is smooth for t ≥ 0. In both cases we have vanishing initial conditions. In the first case, (7.5), we then also want to use the special form of f in (7.1), f = f1 + κ −1 ∂a (κF1a ).
(7.7)
For the lowest order energy estimate it will be useful to take advantage of the special form (7.7) because it will allow us to estimate F with one less space derivative and one more time derivative instead, and in the estimate for the divergence equation we have one more time derivative than space derivatives. Let gs = |g |s+1,∞ ,
hs = |h |s+2,∞ ,
(7.8)
and let K2 denote a continuous function of h0 + g0 + c1 + T −1 + r
(7.9)
which in what follows also depends on the order r of differentiation. We will use that by interpolation, (gs + hs )(gr + hr ) ≤ K2 (gr+s + hr+s ).
(7.10)
Theorem 7.1. Suppose that the initial data in (7.2) vanishes and f in (7.1) or f1 and F1 in (7.7) are smooth and vanish to all orders as t → 0. Suppose also that e = e (h) in (7.1) is a smooth function of h and that h and g are smooth. Then for r ≥ 1 the solution of (7.1) satisfies the estimates ψ 0,r + ψ 1,r−1 ≤ K2
r s=1
(hr−s + gr−s ) 0
t
( f1 0,s−1 + F1 0,s )dτ. (7.11)
350
H. Lindblad
Furthermore ψ r ≤ K2
r
t
(hr−s + gr−s )
( f1 0,s−1 + F1 0,s ) dτ + f s−2
(7.12)
0
s=1
and ψ r ≤ K2
r
t
(hr−s + gr−s )
( f˙ s−2 + f 0 ) dτ,
(7.13)
0
s=1
where f s−2 should be interpreted as 0 if s = 1. Proof. Let us first prove (7.11). Let Er be as in Lemma 7.2 and let E˜ r2 = Er2 +
ψ 2 κdy +
r−1 s=0
gab (Dˆ ts F1a )(Dˆ ts F1b ) κdy.
(7.14)
Then it follows that E˜ r satisfy the same inequality as Er in Lemma 7.2 does, even without the last term in (7.19) since the norm of ψ is included in E˜ 1 : d E˜ r ≤ K2 (hr−s + gr−s ) E˜ s + F1 0,s + f1 0,s−1 . dt r
(7.15)
s=1
The highest order energy in the right hand side E˜ s with s = r can be removed by multiplying with the integrating factor eKt and integrating. We get E˜ r ≤ K2
r
t
(hr−s + gr−s )
( f1 0,s−1 + F1 0,s ) dτ
0
1
+K2
r−1
(hr−s + gr−s )
t
E˜ s dτ.
(7.16)
0
s=1
We claim that if r ≥ 1 , E˜ r ≤ K2
r s=1
(hr−s + gr−s )
t
( f1 0,s−1 + F1 0,s ) dτ.
(7.17)
0
If r = 1, we have just proven it in (7.16) and in general it follows from (7.16) using induction an interpolation. We claim that (7.11) follows from (7.17). In fact Dˆ tr−1 (g ab ∂b ψ) ≤ C E˜ r , and by r−1 ab r−1−s ˇ Lemma 6.3 Dˆ tr−1 (g ab ∂b ψ) = g ab ∂b Dtr−1 ψ+ r−2 gbc )Dˆ ts (g cb ∂b ψ). s=0 s g (Dt r If follows that ∂Dtr−1 ψ ≤ i=1 gr−s E˜ s which together with (7.16) and interpolation prove (7.11). Finally, (7.12) follows t from (7.11) and Lemma 7.3, and (7.13) follows from (7.12) using that f s−2 ≤ 0 f˙ s−2 dτ .
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
351
Lemma 7.2. Suppose that g ab and e = e (h) are smooth and satisfy (7.4) for r ≥ 1, Er (t) = a =
r−1 1
e (Dts+1 ψ)2 + gab (Dˆ ts a )(Dˆ ts b )κdy
1/2
2 s=0 ab g ∂b ψ + F1a .
, (7.18)
Then for r ≥ 1 , dEr (hr−s + gr−s ) Es + F1 0,s + f1 0,s−1 ≤ K2 dt r
s=1
+K(gr−1 + hr−1 ) ψ 0 .
(7.19)
Proof. We will prove that dEr2 /dt is bounded by Er times the right-hand side of (7.19), and (7.19) follows from this since dEr /dt = (dEr2 /dt)/(2Er ). With the notation Dˆ t = Dt + div V and Dˇ t = Dt − div V we have Dˆ t (e g 2 ) = (Dˇ t e )g 2 + 2e g Dˆ t g, for any functions e and g, and since also Dt κ/κ = div V it follows that r−1 dEr2 e (Dts+1 ψ)(Dts+2 ψ) + gab (Dˆ ts a )(Dˆ ts+1 b ) κ dy = dt s=0
r−1 1 ˆ + (Dt e )(Dts+1 ψ)2 + (Dˇ t gab )(Dˆ ts a )(Dˆ ts b ) κ dy. (7.20) 2 s=0
Here the terms on the second row are bounded by K2 Er2 . Therefore it remains to look on the terms on the first row. Since Dˆ t (e ) = κ −1 Dt (κe ) it follows that e
Dts+2 ψ
= Dˆ ts Dt2 (e ψ) +
s+1
Bis Dti ψ,
i=0
Bis =
s
s κ −1 cij (Dt
s−j
j +2−i
κ)(Dt
e ).
(7.21)
j =max(0,i−2)
Furthermore, using Lemma 6.3 we get, since a = ∂a ψ + F a , Dˆ ts+1 a = g ab ∂a Dts+1 ψ + g ab Dts+1 F b s s + 1 ab ˇ s+1−i g (Dt − gbc )Dˆ ti c . i
(7.22)
i=0
= e (h) it follows that the L2 norm of the sums in (7.21) and (7.22) are bounded Since e by K2 rs=0 (gr−s + hr−s )Es , which is included in the right hand side of (7.19) and so is g ab Dts+1 F b . Therefore it remains to consider
352
H. Lindblad
r−1
(Dts+1 ψ)(Dˆ ts Dt2 (e ψ) + Dˆ ts a ∂a Dts+1 ψ) κ dy,
s=0 r−1
(Dts+1 ψ) Dˆ ts Dt2 (e ψ) − κ −1 ∂a κ Dˆ ts a κdy
=
=
s=0 r−1 s=0
(Dts+1 ψ) Dˆ ts f1 )κdy.
(7.23)
Here we have integrated by parts using that Dts+1 ψ ∂ = 0, that Dt2 (e ψ)−κ −1 ∂a κ a = f1 and that Dˆ ts κ −1 ∂a κ a = κ −1 ∂a κ Dˆ ts a . One can get additional space regularity from taking time derivatives of Eq. (7.1) and solving the Dirichlet problem for the Laplacian. Lemma 7.3. If r ≥ 1 then ψ r ≤ K2
r
(gr−i + hr−i )( ψ 0,i + ψ 1,i−1 + f i−2 ),
(7.24)
i=1
where f i−2 is to be interpreted as 0 if i = 1. Proof. We can use the estimates for the Dirichlet problem: ψ = Dt2 (e ψ) − f, ψ ∂ = 0
(7.25)
from Theorem 6.1, since also e = e (h) , ψ r−s,s ≤ K2 (gr−i−j + hr−i−j ) ψ i,j + f i,j −2 , (7.26) i≤r−s−2, j ≤s+2
where f i,j −2 is to be interpreted as 0 if j − 2 < 0. Hence using induction and interpolation we get ψ r ≤
K2
r
(gr−i + hr−i )( ψ 0,i + ψ 1,i−1 + f i−2 ),
(7.27)
i=0
where ψ 1,i−1 is to be interpreted as 0 if i = 0 and f i−2 is to be interpreted as 0 if i − 2 < 0. Theorem 7.4. Suppose that φ is a solution of Dt2 φ − (p φ) = f,
φ t=0 = φ˙ t=0 = 0,
(7.28) where p = p (h) = 1/e (h) and f vanishes to all orders as t → 0; Dtk f t=0 = 0, for k ≥ 0. Let W1 = ∇q, q = φ, q ∂ = 0, (7.29)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
353
and F1 = ∇q ,
q ∂ = 0,
q = f,
(7.30)
Then for r ≥ 1 , W r+1 ≤
K2
r
t
(hr−s + gr−s )
( F˙1 s−1 + F1 0 ) dτ.
(7.31)
0
s=1
Proof. By Theorem 7.1 , φ r ≤ K2
r s=1
t
(hr−s + gr−s )
( F˙1 s−1 + F1 0 ) dτ,
(7.32)
0
which by inverting the Laplacian in (7.29) proves that ∂W1 r is bounded by the right hand side of (7.31) and it only remains to prove the estimate for Dtr+1 W1 . Using (2.54) we can write (7.28) as Dˆ t2 φ − 2σ˙ Dˆ t φ + k φ − (p φ) = f,
k = σ˙ 2 − σ¨ ,
(7.33)
and so div W¨ 1 − ∇ p div W1 −2σ˙ W˙ 1 + k W1 − F1 = −2(∂a σ˙ )W˙ 1a + (∂a k )W1a ,
(7.34)
j j j div Dˆ t W¨ 1 − Dˆ t ∇ p div W1 −Dˆ t 2σ˙ W˙ 1 − k W1 + F1 j = −Dˆ t 2(∂a σ˙ )W˙ 1a − (∂a k )W1a .
(7.35)
and hence
We claim that j −1
P Dˆ t W1 = P Bj (W1 , . . . , Dˆ t j
W1 ), j −1 j ˇ j −i j −1 ˆ Bj (W1 , . . . , Dt W1 ) = − (Dt gab )Dˆ ti W1 . i
(7.36)
i=0
j j −i j j In fact 0 = P Dt ∂a q = P Dt gab W1b = P i=0 ji (Dˇ t gab )Dˆ ti W1 . Furthermore, let j (7.37) q j ∂ = 0. q j = −Dˆ t 2(∂a σ˙ )W˙ 1a − (∂a k )W1a , To say that div H = 0 is equivalent to saying that (I − P )H = 0 so it follows from (7.35)–(7.37) that j j −1 j −2 − ∇ p div W1 Dˆ t W1 = P Bj (W1 , . . . , Dˆ t W1 ) + (I − P )Dˆ t +2σ˙ W˙ 1 − k W1 + F1 + ∇q j −2 ,
(7.38)
354
H. Lindblad
Here j −2 i j −2 i j −2 a j −2−i ab Dˆ t (Dˆ t ∇ p div W1 = g )∂b i m i=0 m=0 × (Dti−m p (h))(Dtm div W1 ) ,
(7.39)
and by interpolation j −2+1 (gj −2−s + hj −3−s ) div W1 s . ∇ a p div W1 ≤ K2
j −2
Dˆ t
(7.40)
s=0
Since the projection P maps L2 to L2 and since inverting (7.37) maps L2 to H 1 (in fact to H 2 ) it therefore follows that Dtr+1 W1 ≤ K2
r
gr−s Dts W1 + (gr−1−s + hr−2−s ) div W1 s
s=0
+K2
r−1
gr−1−s Dts F1 ,
(7.41)
s=0
Equation (7.31) t follows from this if we use (7.32) to estimate φ = div W1 and use that Dts F1 ≤ 0 Dts F˙1 dτ . Corollary 7.5. With assumptions as in Theorem 7.1 we have |ψ |r,∞ ≤ K2 (gr+r0 −1 + hr+r0 −1 ) |f |0,∞ + |f |r+r0 −1,∞ ,
(7.42)
where r0 = [n/2] + 1. Proof. By Sobolev’s lemma ψ r,∞ ≤ C ψ r+r0 , which in turn can be estimated by Theorem 7.1. Then we estimate the L2 norms of f by L∞ norms and use interpolation. Let us now prove that the solution of (7.1) depends smoothly on parameters if the metric g and the inhomogeneous term f do. We have: ∞ ([0, T ] × ) , g ab ∈ C m B k , C ∞ ([0, T ] × Lemma 7.6. Suppose that f ∈ C m B k , C00 ) , where B k = {r ∈ Rk , |r| ≤ ε}. Suppose also that g satisfies the coordinate condition uniformly in B k . Let ψ be the solution of ψ = c2 Dt2 ψ − ψ = f, ψ ∂ = 0 ψ t=0 = Dt ψ t=0 = 0, (7.43) ∞ ([0, T ] × ) . where is given by (7.3). Then ψ ∈ C m B k , C00 Proof. We note that it suffices to prove the statement in the theorem for m = 1, since general case follows from this by induction. In fact, assuming that ψ ∈ the ∞ C l B k , C00 ([0, T ] × ) , l < m then for |α| ≤ l: Drα ψ = Drα f − cβα (γ ) Drβ ψr , (7.44) β+γ =α, |β|<|α|
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
355 γ
where γ = (γj , . . . γ1 ) are ordered multi-indices with γi ∈ {1, . . . , k}. Here Dr = (γ ) Dr γj · · · Dr γ1 , where Dr i = ∂/∂r i , and r are the repeated commutators defined inductively by (i,γ ) = [Dr i , (γ) ], where ( ) = r . The right-hand side of (7.44) is ∞ 1 k then in C B , C00 ([0, T ] × ) , so it follows from the statement in the theorem for ∞ ([0, T ] × ) . m = 1 that ψ ∈ C l+1 B k , C00 Let us write ψr , gr fr , and r = gr , to indicate the dependence of r. First we will ∞ ) and g ∈ C(B k , C ∞ ) implies that ψ ∈ C(B k , C ∞ ). We prove that fr ∈ C(B k , C00 r r 00 will only prove this for r = 0 since the proof in general is just a notational difference from the proof for r = 0. We have r (ψr − ψ0 ) = fr − f0 − ( r − 0 )ψ0 .
(7.45)
∞ ) and tends to 0 in C(B k , C N ), for any N , as r → 0. The right hand side is in C(B k , C00 Since in Corollary 7.5 we have uniform bounds for −1 r , it follows that ψr − ψ0 tends ∞ ). k N to 0 in C(B , C ), for any N , as r → 0, i.e. ψr ∈ C(B k , C00 ∞ 1 k 1 Let us now assume that fr ∈ C (B , C00 ) and gr ∈ C (B k , C ∞ ). Let ψ˙ r = Dr ψr be defined by r ψ˙ r = f˙r − ˙ r ψr , ψ˙ r ∂ = 0, ψ˙ r t=0 = Dt ψ˙ r t=0 = 0, (7.46)
where ˙ r = [Dr , ], f˙r = Dr fr and Dr = (Dr 1 , . . . , Dr k ). Since the right-hand side ∞ ) it follows as above that ψ ˙ r ∈ C(B k , C ∞ ). It remains to prove of (7.46) is in C(B k , C00 00 that ψr is differentiable. We have r ψr − ψ0 − r ψ˙ 0 = fr − f0 − r f˙0 + r − 0 − r ˙ 0 ψ0 +r( r − 0 )ψ˙ 0 . (7.47) Then the right-hand side divided by r tends to 0 in C N , for any N , as r → 0. In view of ˙ the uniform bounds for −1 r in Corollary 7.5 it follows that (ψr − ψ0 − r ψ0 )/r, tends ∞ ). N to 0 in C for any N , as r → 0. Hence ψ ∈ C 1 (B k , C00 8. Tame Estimates for the Divergence Free Equation Let
W t=0 = W˙ t=0 = 0, (8.1) where H is smooth and vanishes to order r as t → 0, i.e. Dtk H t=0 = 0, for k ≤ r. Here i W c , i = 0, 1, are projected multiplication operators. Let Bi W a = P g ab βbc W¨ + AW + B0 W + B1 W˙ = H,
div H = 0,
gs = |g |s+1,∞ ,
hs = |h |s+2,∞ ,
βs = |β |s,∞ + |β |s,∞ , 0
1
ks = gs + hs + βs ,
(8.2)
and let K2 denote a continuous function of k0 + c0−1 + c1 + T −1 + r,
(8.3)
which in what follows also depends on the order r of differentiation. We will use that by interpolation, ks kr ≤ K2 kr+s , for r, s ≥ 0. We will prove the following estimates
356
H. Lindblad
Theorem 8.1. Suppose that (2.4)–(2.5) hold for 0 ≤ t ≤ T and suppose also that x is smooth for 0 ≤ t ≤ T , T ≤ 1. Then (8.1) has a smooth solution for 0 ≤ t ≤ T . It satisfies the estimates t r ˙ kr−s H s dτ, (8.4) W r + W r ≤ K2 0
s=0
for r ≥ 0 and for m = 0, 1, 2, m+1
j Dˆ t W r ≤ K2
j =0
r m
kr+m−j −s
t 0
j Dˆ t H s dτ.
s=0 j =0
(8.5)
Furthermore, for r ≥ 1, r−1 ... W r−1 + W¨ r−1 + W˙ r + W r ≤ K2
t
kr−1−s H¨ s + kr−s H˙ s
s=0 0
+kr+1−s H s + curl H s dτ. (8.6) As pointed out before, existence of smooth solutions for (8.1) was proven in [L3] so we only need to prove the estimates. The theorem with the norms replaced by norms with just space derivatives was proven in [L3]. The theorem here is actually simpler to prove than the one there. First we note that we can reduce to the case B0 = B1 = 0 since the general case follows from this case. Let us show this for (8.4). Assume that (8.4) holds for the case when ¨ + AW = H1 = H + B0 W + B0 = B1 = 0. Then using (8.4) applied to the equation W B1 W˙ gives using that, by Theorem 6.1, Bi W s ≤ K2 sk=0 (βs−k + gs−k ) W k and (8.4), W˙ r + W r ≤ K2
r s=0
(gr−s + hr−s + βr−s )
×
t
( H s + W s + W˙ s ), dτ.
(8.7)
0
We can now first remove the highest order terms from the right-hand side by a Gr¨onwall t type of argument since they occur in the left. In fact let f (t) = 0 W˙ r + W r dτ . Then by (8.7) f ≤ K2 f + K2 H r + r−1 , where r−1 is the sum of the first r − 1 terms in the right of (8.7). Hence multiplying by the integrating factor eK2 t and integrating t gives f (t) ≤ K2 0 H r + r−1 dτ . It follows that W˙ r + W r ≤ K2
r
(gr−s + hr−s + βr−s )
H s dτ
0
s=0
+K2
t
r−1 s=0
t (gr−s + hr−s + βr−s ) ( W s + W˙ s ), dτ. (8.8) 0
This proves (8.4) for r = 0. Equation (8.4) for r ≥ 1 now follows by induction, using (8.4) for s ≤ r − 1 in the terms on the second row of (8.8) together with the interpolation
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
357
(8.4). We can reduce the proof to the case B0 = B1 = 0 also, for (8.5) follows in the same way, using that we have already proven the estimate for smaller m. To prove that (8.6) can be reduced to the case B0 = B1 = 0, we also estimate the lower order terms using (8.5). We now in what follows assume that B0 = B1 = 0. Of course we could have included these operators in the calculations that follow but the argument becomes clearer without them. Lowering the indices in (8.1): GW¨ + AW = GH.
(8.9)
Let LIT , I ∈ T , stand for a product of Lie derivatives of |I | vector fields in T and let WI = Lˆ IT W. If we apply repeatedly Lie derivatives LT and the projection in between, see Sect. 4, we obtain cI1 I2 GI1 W¨ I2 + AI1 WI2 − GI1 HI2 = 0, (8.10) where the sum is over all combinations of I1 + I2 = I and cI1 I2 = 1. Here GI and AI are the operators given by (4.39) and (4.40). If we raise the indices again we get W¨ I + AWI = −c˜I1 I2 AI1 WI2 − c˜I1 I2 GI1 W¨ I2 + cI1 I2 GI1 HI2 ,
(8.11)
where c˜I1 I2 = 1, if |I2 | < |I |, and c˜I1 I2 = 0 if |I2 | = |I |. Let us define energies EI = W˙ I , W˙ I + WI , (A + I )WI ,
EsT =
EI .
(8.12)
|I |≤s,I ∈T
Note that in the sum we also included all time derivatives Lˆ Dt . The reason for this is that when calculating commutators second order time derivatives show up in the first term on the right in (8.10). We get by differentiating (8.11), see the end of Sect. 3, ˙ W˙ I + WI , (A˙ + G)W ˙ I . E˙ I = 2W˙ I , W¨ I + (A + I )WI + W˙ I , G
(8.13)
We now want to estimate the right-hand side by ErT , where r = |I |. Here, by (3.27), the last two terms can be bounded by (h0 c0−1 + g0 )EI . Therefore, (8.12) can be estimated by the L2 norm of the right hand side of (8.10). In the sums with cI1 I2 , in (8.9), we have |I2 | < |I |. Since we included time derivatives up to highest orders in ErT it follows that W¨ I2 ≤ ErT . Here GI1 is a bounded operator so the term with GI1 can be controlled. AI1 is an operator of order one so the term with AI1 can be controlled by a constant times ∂WI2 + WI2 . However, at this point we only have control of tangential derivatives of W . One could combine the estimate here with the curl estimate given later to get control over all derivatives up to order r = |I |. Instead we will add a lower order term to the energy such that its time derivative cancels the terms with AI1 and replaces them with lower order terms that may be controlled. Let DI = 2c˜I1 I2 WI , AI1 WI2 , where the sum is over all I1 + I2 = I , with|I2 | < |I |. Then D˙ I = 2c˜I1 I2 W˙ I , AI1 WI2 + WI , A˙ I1 WI2 + WI , AI1 W˙ I2 .
(8.14)
(8.15)
358
H. Lindblad
Hence ˙ I E˙ I + D˙ I = 2c˜I1 I2 WI , A˙ I1 WI2 + WI , AI1 W˙ I2 + WI , AW +2W˙ I , −c˜I1 I2 GI1 W¨ I2 + cI1 I2 GI1 HI2 ˙ W˙ I + WI , GW ˙ I . +WI + W˙ I , G
(8.16)
Here the terms on the first row can be controlled using (3.7) and the terms on the terms on the second row can be controlled using (3.10). We get |U, AJ W | ≤ |h |s+1,∞ c0−1 U, AU 1/2 W, AW 1/2 , |U, GJ W | ≤ |g |s,∞ U W , |J | = s.
(8.17)
Hence, we have proven that |E˙ I + D˙ I | ≤ CErT |DI | ≤ CErT
r
(hr−s c0−1 + gr−s ) EsT + H Ts ,
s=0 r−1
hr−s c0−1 EsT ,
(8.18)
s=0
so it follows that ErT
≤
C(h0 c0−1 +C
+ g0 )
r−1
0
t
(ErT + H r )dτ
(hr−s c0−1
s=0
t EsT + H Ts dτ + EsT . + gr−s )
(8.19)
0
Using induction and interpolation (8.4) we get the same inequality without EsT , for s ≤ r − 1, but instead multiplied by a constant K2 depending on (8.3) and r. Then, by a Gr¨onwall type of argument, see the beginning of the proof, we can also remove ErT t from the right-hand side of (8.17), i.e., with f (t) = 0 ErT dτ , we get an inequality f ≤ K2 f + K2 rs=0 (hr−s + gr−s ) H Ts an multiplying by the integrating factor eK2 t gives: Lemma 8.2. For r ≥ 0 we have ErT
≤
K2
r s=0
(hr−s + gr−s ) 0
t
H Ts dτ.
(8.20)
This proves the estimates for tangential derivatives. We now also want to have estimate for the curl and then by the results in Sect. 5 the estimates for all derivatives will follow from this. Let wa = gab W b and w˙ a = gab W˙ b and let curl wab = ∂a wb − ∂b wa . Then Dt wa = w˙ a + g˙ ab W b , where g˙ ab = Dˇ t gab and W˙ b = Dˆ t W b . It follows that Dt curl wab = curl w˙ ab +∂a (g˙ bc W c )−∂b (g˙ ac W c ). Since the curl of A vanishes it follows that curl w¨ =
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
359
curl H , if w¨ a = gab W¨ b . Since the curl commutes with the Lie derivative it therefore follows that |Dt curl w| ˙ U r−1 ≤ C |Dt curl w|U r−1 ≤ C
r s=0 r
U gr−s |W˙ |U s + | curl H |r−1 ,
(8.21)
gr−s |W |U s ,
(8.22)
s=0
and by Lemma 5.3: |W |U r ≤K1
r
U T gr−1−s | curl w|U s−1 + | div W |s + |W |s ) + K1 gr−1 |W |0 , (8.23)
s=1
|W˙ |U r ≤K1
r
˙ U ˙ T ˙ gr−1−s | curl w| ˙ U s−1 + | div W |s + |W |s ) + K1 gr−1 |W |0 . (8.24)
s=1
Let ˙ r−1 + curl w r−1 . CrU = curl w
(8.25)
Since div W = div W˙ = 0 it now follows that |C˙ rU | ≤ K1 ≤ K1
r s=0 r
gr−s ( W˙ s + W s ) + C curl H r−1 gr−s (CsU + EsT ) + C curl H r−1 .
(8.26)
s=0
This together with Lemma 8.2 and the argument for its proof gives Lemma 8.3. For r ≥ 0 we have W˙ r + W r + ErT ≤ K2
r
(gr−s + hr−s )
t
H s dτ.
(8.27)
0
s=0
This proves the first part of Theorem 8.1. In order to prove the second part of Theorem 8.1 we replace the energy in (8.10) by
T Er,m =
EI ,
(8.28)
I =I1 +I2 , I1 ∈T , |I1 |≤r, I2 ={Dt ,... ,Dt}, |I2 |=m
i.e. we take two additional time derivatives. Noting that the argument leading up to Lemma 8.2 only requires that we have at least as many time derivatives, as tangential space derivatives, so it follows from its proof that Lemma 8.4. For r ≥ 0 we have T Er,m
≤
K2
m r s=0 j =0
kr+m−j −s
t 0
j Dˆ t H Ts dτ.
(8.29)
360
H. Lindblad
... We have w¨ a = −Aa W + H a and w a = gab Dˆ t W¨ b = Dt w¨ a − (Dˇ t gab )W b so, ... since curl AW = 0, curl w¨ = curl H and curl w ab = curl (Dt H )ab − ∂a Dˇ t gbc )W b + ... ¨ s−1 = | curl H |s−1 and | curl w|s−1 ≤ K1 sj =0 gs−j ∂b (Dˇ t gac )W c . Hence | curl w| |W |j + |Dt curl H |s−1 , so using the estimates (8.23)–(8.23) with (W, W˙ ) replaced by ... (W¨ , W ) gives: |W¨ |U r ≤ K1 ... |W |U r ≤ K1
r s=0 r
gr−1−s |W¨ |Ts + K1
r
gr−1−s | curl H |s−1 ,
(8.30)
s=1 r ... T gr−1−s (|W |U + | | gr−1−s |Dt curl H |s−1 . (8.31) + K W 1 s s
s=0
s=1
Using the estimate in Lemma 8.3, twith r replaced by r − 1, the estimate in Lemma 8.4, the estimate Dtk curl H s−1 ≤ 0 Dtk+1 curl H s−1 dτ , and interpolation gives: Lemma 8.5. For r ≥ 0 and m = 0, 1, 2, we have m+1
j T Dˆ t W r + Er,m ≤ K2
j =0
m r
kr+m−j −s
s=0 j =0
t 0
j Dˆ t H s dτ.
(8.32)
We now want to use the estimate for the additional time derivatives to get estimates for W r and W˙ r in (8.6) since by Lemma 5.5: c0 W r ≤ K2 curl w r−1 + div W r−1 + W Tr−1,A + W˙ Tr−1 +
r−1
gr−1−s W s ,
(8.33)
s=0
˙ r−1 + div W˙ r−1 + W˙ Tr−1,A c0 W˙ r ≤ K2 curl w + W¨ Tr−1 +
r−1
gr−1−s W˙ s ,
(8.34)
ALˆ IT W .
(8.35)
s=0
where W Ts,A =
|I |=s,I ∈T
The terms of order r − 1 or less in (8.33)–(8.34) can be estimated by Lemma 8.3 and Lemma 8.5 with r replaced by r − 1. Therefore it only remains to estimate the terms involving the curl and the operator A. The estimate for AWJ with |J | ≤ r − 1 we get from (8.11): AWJ = −c˜J1 J2 AJ1 WJ2 − cI1 J2 GJ1 W¨ J2 − GI1 HJ2 , (8.36) where c˜J1 J2 = 0 if |J2 | = r − 1. Here the terms in the parenthesis can be estimated by t Lemma 8.5, with r replaced by r − 1. Note that HJ2 ≤ 0 H˙ J2 dτ since we assume that H vanishes to all orders as t → 0. Since by (3.8) AJ1 is order one and |J2 | ≤ r − 2
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
361
in the first term on the right, it follows that also this term can be estimated by Lemma 8.3 with r replaced by r − 1. We hence get W Tr−1,A ≤ K2
r−1 t
kr−1−s H¨ s + kr−s H˙ s + kr+1−s H s dτ. (8.37)
s=0 0
From (8.9) we also get the estimate for AW˙ J with |J | ≤ r − 1 by letting one of the derivatives in (8.9) be a time derivative so I = J + {Dt }. If we write this out we get AW˙ J = −c˜J1 J2 AJ1 W˙ J2 − cJ1 J2 A˙ J1 WJ2 ... ˙ J1 W¨ J2 − GJ1 H˙ I2 − G ˙ J1 HJ2 , −cJ1 J2 GJ1 W J2 + G
(8.38)
where c˜J1 J2 = 0 if |J2 | = r − 1. Here the terms in the parenthesis can be estimated by Lemma 8.5, with r replaced by r − 1. Since AJ1 and A˙ J1 is order 1, see (3.8), we can also estimate all the other by Lemma 8.3 with r replaced by r − 1, apart from the term ˙ J . This term is estimated by AW ˙ J ≤ Ch0 ( W r + W r−1 ), where the last with AW term can again be estimated by Lemma 8.3 with r replaced by r − 1. We hence get W˙ Tr−1,A ≤ h0 W r +K2
r−1 t
kr−1−s H¨ s + kr−s H˙ s + kr+1−s H s dτ. (8.39)
s=0 0
Equations (8.33) and (8.34) together with (8.37) and (8.39) therefore give c0 W r ≤ K2 curl w r−1 r−1 t +K2 kr−1−s H¨ s + kr−s H˙ s + kr+1−s H s dτ. (8.40) s=0 0
and ˙ r−1 c0 W˙ r ≤ h0 c0−1 curl w r−1 + K2 curl w r−1 t kr−1−s H¨ s + kr−s H˙ s + kr+1−s H s dτ. (8.41) +K2 s=0 0
It therefore only remains to control the curl. With CrU = curl w ˙ r−1 + curl w r−1 it hence follows from (8.26) and (8.40)–(8.41) that |C˙ rU | ≤ K2 CrU + K2
r−1 t
kr−1−s H¨ s + kr−s H˙ s
s=0 0
+kr+1−s H s dτ + curl H r−1 ,
(8.42)
t where we also used that, by Lemma 8.3, W˙ 0 + W 0 ≤ K2 0 H 0 dτ . Integrating this equation gives a bound for CrU in terms of the integral in (8.38). Using this bound in (8.40)–(8.41) then gives
362
H. Lindblad
Lemma 8.6. For r ≥ 1 we have W˙ r + W r ≤ K2
r−1 t
kr−1−s H¨ s + kr−s H˙ s + kr+1−s H s
s=0 0
+ curl H s dτ.
(8.43)
This concludes the proof of Theorem 8.1. 9. Existence and Tame Estimates for the Inverse of the Modified Linearized Operator We want to show existence and tame estimates for the inverse of the linearized operator. However, first we will show existence and estimates for the modified linearized operator L1 given by (2.55): W t=0 = W˙ t=0 = 0, (9.1) L1 W = F, where F is smooth and vanishes to all orders as t → 0. Let ns = |g |s+2,∞ + |ω |s+1,∞ + |h |s+3,∞ ,
(9.2)
and let K3 denote a continuous function of n0 + c0−1 + T −1 + c1 + r,
(9.3)
which in what follows also depends on the order of differentiation r. In the proof that follows we will use the interpolation ns nr ≤ K3 ns+r .
(9.4)
Theorem 9.1. Suppose that (2.4)–(2.5) hold for 0 ≤ t ≤ T and suppose also that x is smooth for 0 ≤ t ≤ T and that T ≤ 1. Then (9.1), where F is smooth and vanishing to all orders as t → 0, has a smooth solution for 0 ≤ t ≤ T . It satisfies the estimates t r W˙ r + W r ≤ K3 nr−s F s dτ, r ≥ 1. (9.5) s=1
0
Proof of Theorem 9.1. The proof of existence and the estimate (9.5) for the modified linearized operator uses the orthogonal decomposition of the vector field into its divergence free part and a gradient of a function vanishing on the boundary. The solution will be further divided into four parts and for each of them we use either Theorem 6.1, Theorem 7.1 or Theorem 8.1. This gives us the estimates in Theorem 9.2. These estimates imply the estimate (9.5). Furthermore, the estimates hold for iterates, i.e. given an iterate for W we define δh = (x)W by (9.10) and then define Wij by (9.8)–(9.20) and (9.12)–(9.13). This gives us a new iterate for W . Theorem 9.2 then gives us uniform bounds for the iterates and applied to the equations for differences of iterates gives us convergence, see [L2, L3]. Now, the solution W of L1 W = F
(9.6)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
363
can be obtained as the sum of four terms, see Sect. 3, W = W0 + W1 ,
W1 = W10 + W11 ,
W0 = W00 + W01 ,
(9.7)
where W0 is divergence free and W1ia = g ab ∂b q1i ,
q1i ∂ = 0,
i = 0, 1,
q11 = ϕ,
q10 = −e (h)δh,
(9.8)
and ϕ satisfies an ordinary differential equation: ϕ t=0 = ϕ˙ t=0 = 0,
Dt2 ϕ + div ϕ = div F + div e (h)δh,
(9.9)
where div = Dt2 e(h) + Dt div V and δh satisfies the wave equation Dt2 (e (h)δh) − δh = (∂c h)W c − div B1 W˙ + 2σ˙ div W˙ − div B0 W δh
∂
−(Dt2 e(h) + σ˙ 2 ) div W, ˙ = 0, δht=0 = δh = 0. t=0
(9.10) (9.11)
Here, the divergence free parts satisfy the evolution equation for the normal operator, (3.16), W¨ 00 + AW00 − B10 W˙ 00 − B00 W00 = −AW10 + B11 W˙ 1 + B01 W1 + P F, W00 t=0 = W˙ 00 t=0 = 0. (9.12) and W¨ 01 + AW01 − B10 W˙ 01 − B00 W01 = −AW11 ,
W01 t=0 = W˙ 01 t=0 = 0. (9.13)
If Er =
1
W˙ ij r + Wij r + δh r ,
(9.14)
i,j =0
then we get from Theorem 9.2 that for r ≥ 1 Er ≤
K3
r
t
nr−s
( F s + Es ) dτ.
(9.15)
0
s=1
Using a Gr¨onwall type of argument and induction as in Sect. 8 it follows that for r ≥ 1 we have Er ≤
K3
r s=1
which proves Theorem 9.1.
t
nr−s 0
F s dτ
(9.16)
364
H. Lindblad
Theorem 9.2. If r ≥ 1 we have W˙ 11 r + W11 r ≤ K3 W¨ 11 r ≤ K3
r s=1 r
r
F s + W10 s dτ,
nr−s
nr−s F s + W10 s +
W˙ 01 r + W01 r ≤ K3 W˙ 00 r + W00 r ≤ K3 W˙ 10 r + W10 r+1 ≤ K3
s=1 r s=1 r
r
F s + W10 s dτ ,
0
s=1 r
(9.17)
0
(9.18)
r
nr−s
F s + W10 s dτ,
(9.19)
F s + ∂W10 s dτ,
(9.20)
0
r
nr−s 0
t
nr−s
( W˙ s + W s ) dτ,
(9.21)
0
s=1
where W = W0 + W1 , W1 = W10 + W11 and W0 = W00 + W01 . Proof of Theorem 9.2. Equation (9.17) follows directly from applying Lemma 9.3 below to (9.9). Here we write div F1 = div F + div e (h)δh = div F − div W10 + a (∂a div )W10 and use Theorem 6.1. This also gives the additional estimate (9.18). To prove (9.19) we use the second part of Theorem 8.1 applied to (9.13). Since curl AW11 = 0 and we have an estimate for an additional time derivative in (9.18), (9.19) follows. By the first part of Theorem 8.1: W˙ 00 r + W00 r r r ≤ K3 F s + ∂W10 s + W˙ 1 s + W1 s dτ. (9.22) nr−1−s 0
s=0
Using (9.21) and (9.17) gives (9.20). Note that the sum above starts at s = 0 but since the lower norms are included in the higher norms and since we replaced nr−s−1 by nr−s we can start the sum in (9.20) from s = 1. The estimate (9.21) follows from Theorem 7.1 applied to (9.10). If f denotes the right-hand side of (9.10) then f˙ s−2 ≤ K3
s
ns−1−k W˙ k + W k .
(9.23)
k=0
Since f 0 ≤ K3 ( W 2 + W˙ 1 ) we obtain from (7.13) for r ≥ 2 , δh r ≤
K3
r s=0
t
nr−1−s
( W˙ s + W s ) dτ.
(9.24)
0
For r = 1 we use (7.12), which gives (9.24) also for r = 1 if we write f = f1 + (hc W c ). This proves that δh r is bounded by the right hand side of (9.21). To get the estimate
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
365
also for W10 we use Theorem 6.1 to estimate the solution of the Dirichlet problem in (9.8) using that e (h)δh r ≤ K3 rs=0 h r−s,∞ δh s . This gives that ∂W10 r and W10 r are bounded by the right hand side of (9.21) but it remains to show that Dtr+1 W10 is bounded by the right hand side of (9.21) which follows from Theorem 7.4. Lemma 9.3. Let ϕ be the solution of
ϕ˙ t=0 = ϕ t=0 = 0,
Dt2 ϕ + kϕ = f,
(9.25)
j where Dt f1 t=0 = 0 for j ≥ 0 and k = div = Dt2 e(h) + Dt div V . Then for r ≥ 1 we have
ϕ ˙ r−1 + ϕ r−1 ≤ K3 ϕ ¨ r−1 ≤ K3
r−1 s=0 r−1
t
f s dτ,
(9.26)
t nr−1−s f s + f s dτ .
(9.27)
nr−1−s 0
0
s=0
Furthermore let W1 and F1 be defined by W1 = ∇q,
q = ϕ,
F1 = ∇q
q = f,
q ∂ = 0, q ∂ = 0.
(9.28) (9.29)
Then for r ≥ 1 we have W˙ 1 r + W1 r ≤ K3 W¨ 1 r ≤ K3
r s=0 r
t
F1 s dτ,
nr−s
(9.30)
0
t nr+1−s F1 s + F1 s dτ .
(9.31)
0
s=0
Proof of Lemma 9.3 Equation (9.25) is an ordinary differential equation and (9.26)– (9.27) follows as in the proof of Proposition 10.1 in [L3]. Equation (9.25) can be written Dˆ t2 ϕ − 2σ˙ Dˆ t ϕ + k ϕ = f, and so
k = σ˙ 2 + Dt2 e(h),
div W¨ 1 − 2σ˙ W˙ 1 + k W1 − F1 = −2(∂a σ˙ )W˙ 1a + (∂a k )W1a ,
(9.32)
(9.33)
and hence j j j div Dˆ t W¨ 1 − Dˆ t 2σ˙ W˙ 1 − k W1 + F1 = −Dˆ t 2(∂a σ˙ )W˙ 1a − (∂a k )W1a . (9.34) We claim that j j −1 P Dˆ t W1 = P Bj (W1 , . . . , Dˆ t W1 ), j −1 j ˇ j −i j −1 Bj (W1 , . . . , Dˆ t W1 ) = (Dt gab )Dˆ ti W1 . i i=0
(9.35)
366
H. Lindblad
j j −i j j In fact 0 = P Dt ∂a q = P Dt gab W1b = P i=0 ji (Dˇ t gab )Dˆ ti W1 . Furthermore, let j q j = −Dˆ t 2(∂a σ˙ )W˙ 1a − (∂a k )W1a , (9.36) q j ∂ = 0. To say that div H = 0 is equivalent to saying that (I − P )H = 0 so it follows from (9.34)–(9.36) that j j −1 Dˆ t W1 = P Bj (W1 , . . . , Dˆ t W1 ) j −2 +(I − P )Dˆ t 2σ˙ W˙ 1 − k W1 + F1 + ∇q j −2 .
(9.37)
Since the projection P maps L2 to L2 and since inverting (9.36) maps L2 to H 1 (in fact to H 2 ) it therefore follows that: Dtr+1 W1
≤
K3
r
nr−s Dts W1 +
s=0
r−1
nr−2−s Dts F1 .
(9.38)
s=0
Since it also follows from (9.26) that ∂ W˙ 1 r−1 + ∂W1 r−1 ≤ K3 ∂ W¨ 1 r−1 ≤ K3
r s=0 r
t
F1 s dτ,
nr−s
(9.39)
0
t nr−s F1 s + F1 s dτ ,
the lemma follows from also estimating Dts F1 (t, ·) ≤
(9.40)
0
s=0
t 0
Dts+1 F1 (τ, ·) dτ .
10. Estimates for the Enthalpy in Terms of the Coordinate We have now proved that the linearized operator is invertible. However, since we think of h = (x) as a functional of x we must also estimate the L∞ norms of h in terms of the L∞ norms of x. For the corresponding problem for the incompressible case in [L3] we could take advantage of the Schauder estimates. However for the wave equation there are no L∞ estimates that do not lose regularity. For wave equations it is best to get the L∞ norms from the L2 norms using Sobolev’s lemma. These estimates were obtained in Corollary 7.5. However, in the estimates there the L∞ norm also occurred in the right hand side, due to that we assumed that e = e (h). This estimate can easily be improved by estimating the L2 norm of the solution of the nonlinear wave equation instead. We will however for simplicity assume that e(h) = ch, so that e (h) = c, where 0 < c < ∞ is a constant. In that case there are no h terms in the results in Sect. 7. We have Dt2 ch − h = (∂i V j )(∂j V i ), h∂ = 0. (10.1) The estimates in Corollary 7.5 were however formulated for vanishing initial data. Therefore let h˜ = h − h0 , and where h0 satisfies Eq. (10.1) to all orders as t → 0, if x − x0 vanishes to all orders as t → 0, see Sect. 2. It follows that ˜ − h˜ = −Dt2 (ch0 ) + h0 − (∂i V j )(∂j V i ) Dt2 (ch)
(10.2)
vanishes to all orders as t → 0. We can therefore apply Corollary 7.5, which gives:
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
367
Lemma 10.1. We have |g |s+1,∞ + |ω |s,∞ ≤ K1 |x |s+2,∞ .
(10.3)
Suppose that e(h) = ch, where 0 < c < ∞. Then for r ≥ 2 , |h |r,∞ ≤ K2 |x |r+r0 +1,∞ .
(10.4)
Here Ki are as in Definition 5.2 and K2 also depends on a bound for |h0 |r+r0 +1,∞ . Proof. The first inequality follows directly from the definitions and interpolation. If f denotes the right hand side of (10.2) then (10.5) |f |s,∞ ≤ K2 |h0 |s+2,∞ + |x |s+2,∞ . As pointed out above if e(h) = ch, then hr in Sect. 7 vanishes and by Corollary 7.5 and interpolation we have ˜ r,∞ ≤ K2 |x |r+r0 +1,∞ + |h0 |r+r0 +1,∞ . |h |
(10.6)
However since we are just looking on fixed initial data we can also include the norms of h0 in the constants and since in fact we also have a lower bound for |x |1,∞ ≥ Cc1 > 0, by the coordinate condition, the lemma follows. Remark. Note that K2 in (10.4) depends on h0 . However, h0 is a function which is fixed once we fixed the initial data so this just leads to an r dependence of the constant. It now follows that with Ki as in Definition 5.2 and Ki as in Sects. 7, 8, and 9 we have Ki ≤ Ki+r0 +1 .
(10.7)
11. Estimates for the Physical Condition and Coordinate Condition We assume the physical condition and the coordinate condition initially at time 0 for some constants c0 > 0 and c1 < ∞ and we need to show that this implies that they will hold with c0 replaced by c0 /2 and c1 replaced by 2c1 , for 0 ≤ t ≤ T , if T is sufficiently small. Now for the coordinate condition this is easy since we can just estimate the physical condition by the time derivative of g, which can be estimated by |x |2,∞ : Lemma 11.1. Let M(t) = supy∈ M(t) ≤ 2M(0),
for
|∂x/∂y|2 + |∂y/∂x|2 . Then
t ≤ T,
if
T |x |2,∞ M(0) ≤ 1/8. (11.1)
Let N(t) = supy∈∂ |∇N h|−1 . Then assuming that T is so small that (11.1) hold we have N(t) ≤ 2N (0)
for
t ≤ T,
if
T |h |2,∞ M(0)N (0) ≤ 1/8.
(11.2)
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Proof. We have |Dt ∂x/∂y| ≤ |x |2,∞ and |Dt ∂y/∂x| ≤ |∂y/∂x|2 |Dt ∂x/∂y| so M (t) ≤ (1 + M 2 ) |x |2,∞ ≤ 2M 2 |x |2,∞ , since also M(t) ≥ 1. Hence −1 , M(t) ≤ M(0) 1 − 2 |x |2,∞ M(0)t
when
2 |x |2,∞ M(0)t < 1. (11.3)
Now, ∇N h = N a ∂a h, where N is the unit normal, so Dt ∇N h = ∇N Dt h+(Dt N a )∂a h = ∇N Dt h + (Dt N a )gab N b ∇N h, since h∂ = 0. Furthermore 0 = Dt (gab N a N b ) = 2gab (Dt N a )N b+ (Dt gab )N a N b and N a = (∂y a /∂x i )N i , where δij N i N j = 1. Hence |Dt ∇N h| ≤ M |∂Dt h| + |∂Dt x||∇N h| Therefore if N (t) = supy∈∂ |∇N h|−1 , we have N ≤ M |h |2,∞ N 2 + M |x |2,∞ N/2 and if we use (11.1) and multiply with the integrating factor, N˜ (t) = N (t)e−tM(0) |x |2,∞ , we get N˜ ≤ 2e1/8 M(0) |h |2,∞ N˜ 2 . Hence −1 , when N(t) ≤ N(0)e1/8 1 − N (0)2e1/8 M(0) |h |2,∞ t N(0)2e1/8 M(0) |h |2,∞ t < 1. This proves the lemma.
(11.4)
To satisfy the condition in (11.1) we just need to choose T so small that T |x |2,∞ c1 ≤ 1/8. We remark that x = u + x0 , where x0 is fixed and that in the Nash-Moser iteration we will only apply our estimates to functions satisfying |u |r0 +4,∞ ≤ 1. However for the physical condition this is a bit more difficult. One has to control the |h |2,∞ and the estimate for h in terms of x using interpolation so they are in terms of a constant K that is a continuous function of T −1 and tends to infinity as T → 0. In the compressible case we never used interpolation in time so the corresponding estimate there was easier. We therefore have to redo the estimates for the wave equation for the lowest norms without using interpolation in time. This however, follows from standard estimates for the wave equation. Those we have here also work if we do not use interpolation. If we do not use interpolation, then the estimates in Sects. 5,6 and 7 still hold, with constants independent of T ≤ 1, but instead of depending linearly on the highest norms of x they are polynomials in the highest norms of x occurring in the estimates. Lemma 11.2. There are continuous increasing functions Cr such that for T ≤ 1 we have |h |r,∞ ≤ Cr c1 , |x |r+r0 +1,∞ , |h0 |r+r0 +1,∞ .
(11.5)
Proof. The proof is the same as the proof of Lemma 10.1 using that the estimates in Sects. 5,6 and 7 hold with constant of the form above. Summing up, we have hence proven that Lemma 11.3. Let C2 be as in Lemma 11.2 and let c1 and c0 be constants such that the coordinate condition (2.13) and the physical condition (2.14) hold when t = 0. Suppose 0 < T ≤ 1 is fixed such that T c1 (1 + |x0 |2,∞ ) ≤ 1/8, T c1 C2 2c1 , 1 + |x0 |r0 +3,∞ , |h0 |r0 +3,∞ ≤ c0 /8,
(11.6)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
369
where C2 is as in Lemma 11.2. Then for 0 ≤ t ≤ T , the coordinate condition hold with c1 replaced by 2c1 and c0 replaced by c0 /2 if |u |r0 +4,∞ ≤ 1,
(11.7)
where r0 = [n/2]+1 is the Sobolev exponent. Here u = x −x0 and x0 is the approximate solution. Proof. It follows from Lemma 11.2 that |h |2,∞ ≤ C2 c1 , |u + x0 |r0 +3,∞ , |h0 |r0 +3,∞ .
(11.8)
In view of (11.2), the physical condition with c0 replaced by c0 /2 holds if T is so small that (11.5) holds and T c1 C2 2c1 , |u + x0 |r0 +3,∞ , |h0 |r0 +3,∞ ≤ c0 /8.
(11.9)
We recall again that in the Nash-Moser iteration we will only consider u for which (11.7) holds. From now on we will therefore assume that 0 < T ≤ 1 is fixed and so small that (11.6) hold.
12. Tame Estimates for the Inverse of the Linearized Operator in Terms of the Coordinate We have Theorem 12.1. Suppose that T > 0 is so small that the conditions in Lemma 11.3 hold. Suppose also that x = u + x0 and δ are smooth in [0, T ] × and that δ and u vanish to infinite order as t → 0. Then there are constants K, depending on the approximate solution (x0 , h0 ), on (c0 , c1 ) and on r, such that there is a smooth solution δx of ˙ (x)δx = δ, in [0, T ] × δx t=0 = δx = 0 (12.1) t=0 satisfying δ x ˙ r + δx r ≤ K
r
|x |r+r0 +4−s,∞
s=1
t
δ s dτ
(12.2)
0
for r ≥ 1 if |u |r0 +4,∞ ≤ 1.
(12.3)
|x |r+r0 +4,∞ ≤ K + |u |r+r0 +4,∞ .
(12.4)
Moreover
370
H. Lindblad
Proof. First show existence of the equation where the vector field is expressed in the Lagrangian frame, W a = (∂y a /∂x i )δx i and F a = (∂y a /∂x i )δi : L0 W = F, in [0, T ] × W t=0 = W˙ t=0 = 0 (12.5) and that it satisfies the estimate W˙ r + W r ≤ K
r
|x |r+r0 +4−s,∞
s=1
t
F s dτ
(12.6)
0
for r ≥ 1. We have already proved this for L1 W = F in Theorem 9.1, using Lemma 10.1 and Lemma 11.3. Therefore it remains to prove the resultfor L0 W = L1 W −B3 W = F , s where s B3 is given by (2.63). We have B3 W s ≤ K k=0 |x |s+r0 +3−k,∞ W k ≤ K k=1 |x |s+r0 +4−k,∞ W k , if s ≥ 1. Applying the theorem to the equation L1 W = F + B3 W and using interpolation we get that for r ≥ 1 , W˙ r + W r ≤ K
r s=1
|x |r+r0 +4−s,∞
t
( F s + W s ) dτ.
(12.7)
0
If we put up an iteration L1 W k+1 = F − B3 W k , for k ≥ 0 and W 0 = 0 then (12.6) is going to be true with W in the right hand side replaced by W (k) and in the left by W k+1 . It is easiest to first show convergence to a smooth solution and then afterwards prove the estimate (12.2). To show convergence we can just consider the estimate (12.6) where we include the norms of x in the constants and estimate the lower order norms by higher order 1 k+1 − W k . Then L W ˜ k+1 = −B3 W˜ k , for k ≥ 1. norms. Let W˜ k+1 1 ˜ = F and L1 W t = W Hence if Erk = kj =1 Dˆ t W˜ j r + W˜ j r we have Erk+1 ≤ Cr 0 ( F r +Erk ) dτ . Using t a Gr¨onwall type of argument one therefore get uniform bounds Erk ≤ Cr 0 F r dτ , for 0 ≤ t ≤ T ≤ 1. This proves convergence to a smooth solution, W . Once we have a smooth solution it will satisfy the estimate (12.7). By a Gr¨onwall type of argument and induction as in Sect. 8 it follows that the solution also satisfies the estimate (12.6), for some other constant K. Finally, we want to deduce the estimate for δx and δ. The estimate (12.6) is in terms of W a = δxi ∂y a /∂xi and F a = δi ∂y a /∂x i , turning them into an estimate for δx and δ just produces lower order terms of the same form: δ x ˙ r + δx r ≤ K( W˙ r + W r + |x |r+2,∞ W 0 , F r ≤ K( δ r + |x |r+1,∞ δ 0 , (12.8) where inequality in Lemma 5.7. By (12.6) W 0 ≤ W 1 ≤ t we used the interpolation t K 0 F 1 dτ ≤ K 0 δ 1 dτ and by interpolation |x |r+r0 +4−s,∞ |x |s+1,∞ ≤ K |x |r+r0 +3,∞ , Hence (12.3) follows.
1 ≤ s ≤ r.
(12.9)
If we also use Sobolev’s lemma, r0 = [n/2] + 1, and estimate the integrals by the L∞ norms and use interpolation and (12.2) we get |δx |r,∞ ≤ K δ r+r0 ,∞ + |x |r+2r0 +4,∞ δ |0,∞ , r ≥ 0. (12.10)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
371
Furthermore we want to turn it into estimates for ˜ (u) = (u + x0 ) − (x0 )
(12.11)
˜ (u) = (u + x0 ). Let ψ(u) denote the right inverse of ˜ (u). Then see (2.23). Then since |x |r,∞ ≤ |u |r,∞ + |x0 |r,∞ we can again include |x0 |r,∞ in the constants. Hence we have proven that Theorem 12.2. Suppose that T > 0 is so small that the conditions in Lemma 11.3 hold. Suppose also that x = u + x0 and g˜ are smooth in [0, T ] × and that g˜ and u vanish to infinite order as t → 0. Then there are constants K, depending on the approximate ˜ (u) has a solution (x0 , h0 ), on (c0 , c1 ) and on r, such that the linearized operator right inverse ψ(u) satisfying |ψ(u)g | ˜ r,∞ ≤ K |g | ˜ r+r0 ,∞ + |u |r+2r0 +4,∞ g | ˜ 0,∞ , r ≥ 0. (12.12) if |u |r0 +4,∞ ≤ 1.
(12.13)
13. Tame Estimate for the Second Variational Derivative We now first want to show that the Euler map (x) given by (2.20)–(2.21) is C k , i.e. that (x) depends smoothly on k parameters if x does. To be more precise, with ∞ ) if B k = {r ∈ Rk ; |r| ≤ 1}, we want to show that (x) − (x0 ) ∈ C ∞ (B k , C00 ∞ ∞ ∞ k x − x0 ∈ C (B , C00 ), where x0 is the approximate solution satisfying (2.18) and C00 2 is given by (2.19). That Dt and ∂i in (2.20) depends smoothly on parameters is obvious so we only need to prove that h = (x), given by (2.21) does. Subtracting off the approx∞ ) if imate solution h0 of (2.21) we get (10.2), where the right-hand side is in C k (B k , C00 ∞ ∞ ). k k k k x − x0 ∈ C (B , C00 ). Hence it follows from Lemma 7.6 that h − h0 ∈ C (B , C00 We must now also obtain tame estimates for the second variational derivative. If x depends smoothly on the parameter r then the variational derivative of x is δx(t, y) = ∂x(r, t, y)/∂r r=0 , we can e.g. take x = x(t, y) + r δx(t, y). The first variational derivative (x) of the Euler map is given by (x)δxi = δ(x)i = ∂(x)i /∂r r=0 = Dt2 δxi − ∂k h ∂i δx k + ∂i h (δx), (13.1) where δh = h (δx) = (x)δx satisfies Dt2 (cδh) − δh = −δ p − ∂k p δx k − 2(∂i ∂k p)∂ i δx k , and δp
δ h = 2∂k V i ∂i δx l ∂l V k − 2∂k V i ∂i δv k
(13.2)
= 0, where δv = Dt δx. Now, let x depend smoothly on two parameters r and s, such that ∂ 2 x/∂r∂s = 0, and also set x = ∂x/∂s s=0 , e.g. x = x(t, y) + rδx(t, y) + s x(t, y). Then the second variational derivative is given by (x)(δx, x)i = δ(x)i = ∂ ∂i (x)/∂r r=0 /∂s s=0 . (13.3) ∂
We have
372
H. Lindblad
Lemma 13.1. ( x, δx)i = ∂k h ∂i x l ∂l δx k + ∂i δx l ∂l x k −∂k h ( x) ∂i δx k − ∂k h (δx) ∂i x k + ∂i h ( x, δx), (13.4) where h = (x) and h = (x). The estimates for h = (x) and h = (x) must also be obtained: Lemma 13.2. Let h = (x) and let δh = h (δx) = (x)δx be the variational derivative. We have |δh |r,∞ ≤ K |δx |r+r0 +1,∞ + |x |r+2r0 +3,∞ |δx |0,∞ (13.5) and with h (δx, x) = (x)(δx, εx) the second variational derivative, we have h (δx, x) r,∞ ≤ K |δx |r+3r0 +6,∞ | x |0,∞ + | x |r+3r0 +6,∞ |δx |0,∞ + |x |r+3r0 +6,∞ |δx |0,∞ | x |0,∞ .
(13.6)
Proof of Lemma 9.3 The proof is similar to the proof of Lemma 10.1. We have Dt2 (ch) − h = (∂i V j )(∂j V i ), h = κ −1 ∂a κg ab ∂b h . (13.7) It follows that Dt2 (cδh) − δh = δ (∂i V j )(∂j V i ) + κ −1 ∂a δ(κg ab )∂b h − div δx κ −1 ∂a κg ab ∂b h ,
(13.8)
since δκ = κ div δx, see [L1]. Using the estimate for h in terms of x in Lemma 10.1 and Corollary 7.5, if f denotes the right-hand side of (13.8), |f |r,∞ ≤ K |δx |r+2,∞ + |x |r+r0 +3,∞ |δx |0,∞ , (13.9) and hence by Corollary 7.5 |δh |r,∞ ≤ K |δx |r+r0 +1,∞ + |x |r+2r0 +3,∞ |δx |0,∞ .
(13.10)
In the proof we use the interpolation inequalities in Lemma 5.7 and the fact that |x |r0 +4,∞ ≤ K. To calculate the second variation we apply to this, where we assumed that δx = 0. Note that div δx = −(∂i x k )(∂k δx i ), Dt2 (c δh) − δh = δ (∂i V j )(∂j V i ) − div x κ −1 ∂a δ(κg ab )∂b h − div δx κ −1 ∂a (κg ab )∂b h + div x div δx + (∂i x k )∂k δx i ) κ −1 ∂a κg ab ∂b h − div δx κ −1 ∂a (κg ab )∂b h + κ −1 ∂a δ(κg ab ) ∂b h +κ −1 ∂a δ(κg ab )∂b h − div δx κ −1 ∂a κg ab ∂b h +κ −1 ∂a (κg ab )∂b δh − div x κ −1 ∂a κg ab ∂b δh . (13.11)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
373
Here δg ab = −g ac g bd δgcd , δgab = δij (∂a δx i )(∂b x j )+δij (∂a x i )(∂b δx j ) so since δx = 0, we have δgab = 2δij (∂a δx i )(∂b x j ). The first terms on the right of (13.11) gives rise to a term of the form (∂ v)(∂δv), (∂ v)(∂δx)(∂v),
(∂δv)(∂ x)(∂v), (∂ x)(∂δx)(∂v)(∂v) (13.12)
multiplied by powers of ∂y/∂x or ∂x/∂y. If f1 denotes any of these terms then |f1 |r,∞ ≤ K |δx |r+4,∞ | x |0,∞ + | x |r+4,∞ |δx |0,∞ (13.13) + |x |r+5,∞ |δx |0,∞ | x |0,∞ . The terms on the second and third row in (13.11) gives rise to terms of the form (∂ 2 δx)(∂ x)(∂h), (∂δx)(∂ 2 x)(∂h), (∂δx)(∂ x)(∂h)(∂ 2 x)
(∂δx)(∂ x)(∂ 2 h), (13.14)
multiplied by powers of ∂y/∂x or ∂x/∂y. These can be estimated by (13.13) plus K |h |r+4,∞ |δx |0,∞ | x |0,∞≤K |x |r+r0 +5,∞ |δx |0,∞ | x |0,∞ (13.15) by Lemma 10.1. The terms on the last row in (13.11) gives rise to terms of the form (∂ 2 δx)(∂ h), (∂ 2 x)(∂δh), (∂δx)(∂ h)(∂ 2 x)
(∂δx)(∂ 2 h),
(∂ x)(∂ 2 δh),
(∂ x)(∂δh)(∂ 2 x), (13.14)
multiplied by powers of ∂y/∂x or ∂x/∂y. These can be estimated by K |δx |r+3,∞ | h |0,∞ + | x |r+3,∞ |δh |0,∞ + |δh |r+3,∞ | x |0,∞ + | h |r+3,∞ |δx |0,∞
+K |x |r+4,∞ ( |δx |0,∞ | h |0,∞ + |δh |0,∞ | x |0,∞ . (13.15)
If we also use (13.5) we see that this can be estimated by K |δx |r+3,∞ | x |r0 +1,∞ + | x |r+3,∞ |δx |r0 +1,∞ + |δx |r+r0 +4,∞ | x |0,∞ + | x |r+r0 +4,∞ |δx |0,∞
+K |x |r+4,∞ ( |δx |0,∞ | x |r0 +1,∞ + |δx |r0 +1,∞ | x |0,∞ +K |x |2r0 +3,∞ |δx |r+3,∞ | x |0,∞ + | x |r+3,∞ |δx |0,∞ +K |x |r+2r0 +6,∞ |δx |0,∞ | x |0,∞ + | x |0,∞ |δx |0,∞
+K |x |r+4,∞ |x |2r0 +3,∞ ( |δx |0,∞ | x |0,∞ + |δx |0,∞ | x |0,∞ . (13.16)
Using interpolation again this is bounded by K |δx |r+r0 +4,∞ | x |0,∞ + | x |r+r0 +4,∞ |δx |0,∞ + |x |r+2r0 +6,∞ |δx |0,∞ | x |0,∞ .
(13.17)
It follows that |Dt2 (c δh) − δh |r,∞ ≤ K |δx |r+r0 +4,∞ | x |0,∞ + | x |r+r0 +4,∞ |δx |0,∞ + |x |r+2r0 +6,∞ |δx |0,∞ | x |0,∞ .
(13.18)
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H. Lindblad
Hence by Corollary 7.5 : | δh |r,∞ ≤ K |δx |r+2r0 +3,∞ | x |0,∞ + | x |r+2r0 +3,∞ |δx |0,∞ + |x |r+3r0 +5,∞ |δx |0,∞ | x |0,∞ .
(13.19)
Theorem 13.3. Suppose that T > 0 is so small that the assumptions in Lemma 11.3 hold. Then there are constants K depending on the approximate solution (x0 , h0 ), on (c0 , c1 ) and on r such that | (u + x0 )( x, δx) |r,∞ ≤ K |δx |r+2r0 +4,∞ | x |0,∞ + | x |r+2r0 +4,∞ |δx |0,∞ + |u |r+3r0 +6,∞ |δx |0,∞ | x |0,∞
(13.20)
if |u |r0 +4,∞ ≤ 1.
(13.21)
14. The Smoothing Operators We will work in H¨older spaces since the standard proofs of the Nash-Moser theorem uses H¨older spaces. The H¨older norms for functions defined on a compact convex set B ∈ R1+n are given by, if k < a ≤ k + 1, where k ≥ 0 is an integer, |u |a,∞ = |u |H a =
sup
|∂ α u(t, y) − ∂ α u(s, z)| |(t, y) − (s, z)|a−k
(t,y),(s,z)∈B |α|=k
+ sup |u(t, y)|,
(14.1)
(t,y)∈B
and |u |H 0 = sup(t,y)∈B |u(t, y)|. Since we use the same notation for the C k norms, |u |k,∞ we will indicate the difference by simply using letters a, b, c, d, e, f , etc. for the H¨older norms and i, j, k, l, .. for the C k norms. Since a Lipschitz continuous function is differentiable almost everywhere and the norm of the derivative at these points is bounded by the Lipschitz constant, we conclude that for integer values this is the same if the L∞ norm of ∂ α u for |α| ≤ k, and furthermore, since all our functions are smooth it is the same as the supremum norm. Our tame estimates for the inverse of the linearized operator and the second variational derivative are only for C k norms with integer exponents. However, since |u |k,∞ ≤ C |u |a,∞ ≤ C |u |k+1,∞ , if k ≤ a ≤ k + 1, see (14.2), they also hold for non integer values with a loss of of one more derivative. In [L3] we used smoothing only in the space directions but here we will use smoothing also in the time direction. Therefore we define the H¨older space time norms as above. They satisfy |u |a,∞ ≤ C |u |b,∞ ,
a≤b
(14.2)
and they also satisfy the interpolation inequality |u |c,∞ ≤ C |u |λa,∞ |u |1−λ b,∞ ,
(14.3)
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375
where 0 ≤ a ≤ c ≤ b, 0 ≤ λ ≤ 1 and c = λa + (1 − λ)b. Furthermore, the H¨older spaces are rings: |uv |a,∞ ≤ C( |u |a,∞ |v |0,∞ + |u |0,∞ |v |a,∞ ).
(14.4)
For the Nash-Moser technique, apart from tame estimates one also needs a smoothing operator Sθ that satisfies the following properties with respect to the H¨older norms: Lemma 14.1. Let |u | a denote the H¨older norms in (14.1) with B = [0, T ] × , where ∞ = C ∞ [0, T ] × be as in (14.11). Then there is a family of smoothing T ≤ 1. Let C00 00 ∞ → C ∞ , 1 ≤ θ < ∞ such that operators Sθ : C00 00 |Sθ u |a,∞ ≤ C |u |b,∞ ,
a ≤ b,
(14.5)
|Sθ u |a,∞ ≤ Cθ
a−b
|u |b,∞ ,
a ≥ b,
(14.6)
|(I − Sθ )u |a ≤ Cθ
a−b
|u |b,∞ ,
a ≤ b,
(14.7)
|(S2θ − Sθ )u |a,∞ ≤ Cθ
a−b
|u |b,∞ ,
a ≥ 0,
(14.8)
where the constants C only depend on the dimension and an upper bound for a and b. The last property, (14.8) follows from (14.6) for a ≥ b and from (14.7) for a ≤ b. Alternatively, it follows from the following stronger property: d Sθ u |a,∞ ≤ Cθ a−b−1 |u |b,∞ , | dθ
a ≥ 0.
(14.9)
For functions supported in the interior of a compact set K there there are smoothing operators, see [H1], that satisfy the above properties (14.5)–(14.9), with respect to the H¨older norms. These are constructed as follows. Let the Fourier transform φˆ ∈ C0∞ be 1 in a neighborhood of the origin and set φθ (z) = θ 1+n φ(θz), and Sθ u = χ φθ ∗ u, where χ ∈ C0∞ is 1 on a neighborhood of K. However we have functions defined on the compact set [0, T ] × that do not have compact support in . Therefore we need to extend these functions to have compact support in some larger set, without increasing the H¨older norms with more than with a multiplicative constant. There is a standard extension operator in [S] that turns out to have these properties, see Lemma 14.2 below. First however, we note that we will only apply the smoothing operators to functions that vanish to all orders as t → 0. Hence we can extend these functions to be 0 for t ≤ 0 without changing the H¨older norm. Then we extend the functions defined for t ≤ T to functions supported in [0, 2], using the extension in Lemma 14.2, for y ∈ fixed, noting that H¨older continuity in (t, y) follows from differentiability and H¨older continuity in each direction using the triangle inequality and the linearity of the extension operator in Lemma 14.2. Then we want to extend the functions defined in = {y; |y| ≤ 1} to functions supported in {y; |y| ≤ 2}. In order to do this we first remove a region around the origin and introduce polar coordinates r ≥ 0 and ω ∈ S n−1 , which is a nonsingular change of variables away from the origin. Then we use the extension operator in Lemma 14.2 for fixed t and ω, to extend in the radial direction from a function defined for r ≤ 1 to functions supported in r ≤ 2. By the remark above, H¨older continuity in (t, r, ω) follows from H¨older continuity in each direction. Doing the extensions above we hence obtain an extension u˜ of u defined in [0, T ]× such that ˜ ∈ {(t, y); 0 ≤ t ≤ 2, |y| ≤ 2} (14.10) |u | ˜ a,∞ ≤ C |u |a,∞ , a ≥ 0 supp (u)
376
H. Lindblad
for u in ∞ ∞ C00 [0, T ] × = {u ∈ C ∞ ([0, T ] × ), Dtk ut=0 = 0, k ≥ 0}. (14.11) = C00 We note that, in fact the constant in (14.10) is independent of T . Once we have the extension operator we can use the smoothing operators in [H1, H2], defined for compactly supported functions, applied to the extension of our function. Let us call the smoothing operators defined in [H1, H2] S˜θ . These satisfy the properties (14.5)–(14.9). By (14.10) the smoothing operators Sˆθ u given by the restriction of S˜θ u˜ to [0, T ] × then also satisfy the properties (14.5)–(14.9) if u is in (14.11). However, Sˆθ u is not in (14.11) anymore. In our estimates we will only apply the smoothing operators to functions that vanish to all orders as t → 0 and in our estimates we need also Sθ u to vanish to all orders as t → 0. We therefore have to modify our smoothing operators so that this is true. Let χ (t) ∈ C ∞ be a function such that χ (t) = 0, when t ≤ 0 and χ(t) = 1, when t ≥ 1, and let χθ (t) = χ (θ t). Then Sθ u = χθ S˜θ (u) ˜ [0,T ]× (14.12) is in (14.11) and we claim that for functions u in (14.11), (14.5)–(14.9) hold. This follows from Lemma 14.3 below, since the smoothing operators defined in [H1, H2] are convolution operators of the form in Lemma 14.3. Lemma 14.2. There is a linear extension operator Ext : H a ((−∞, 0]) → H a ((−∞, +∞)), where H a are the H¨older spaces, such that Ext(f ) = f when r ≤ 0, and Ext(f ) a,∞ ≤ C f a,∞ .
(14.13)
Here C is bounded when a is bounded. Furthermore, if r ≥ −c in the support of f , where c > 0, then r ≤ c in the support of Ext(f ). Proof. Let Ext(f )(r) = f˜(r), where f˜(r) = f (r), when r ≤ 0, and ∞ ˜ f (r) = f (r − 2λr) ψ1 (λ) dλ, r > 0,
(14.14)
1
where ψ1 is a continuous function on [1, ∞), such that ∞ ∞ ψ1 (λ) dλ = 1, λk ψ1 (λ) dλ = 0, 1
k > 0,
1
|ψ1 (λ)| ≤ CN (1 + λ)−N ,
N ≥ 0.
(14.15)
The existence of such a function was proved in [S] where the extension operator was also introduced. In [S] it was proven that this operator is continuous on the Sobolev spaces but it was not proven there that it is continuous on the H¨older spaces so we must prove this. First we note that if f ∈ C k then the extension is in C k . In fact ∞ f˜(j ) (r) = f (j ) (r − 2λr)(1 − 2λ)j ψ1 (λ) dλ, r > 0. (14.16) 1
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
377
j From the continuity of ∂r f and (14.14)–(14.15) it follows that limr→+0 f˜(j ) (r) = (j ) k f (0), that f˜ is in C , and that f˜ k,∞ ≤ Ck f k,∞ , if k is an integer. Suppose now that k < a ≤ k + 1 where k is an integer. We must now estimate
sup r,ρ
|f˜(k) (r) − f˜(k) (ρ)| |r − ρ|a−k
(14.17)
by Ck ∂rk f a−k,∞ . If r ≤ 0 and ρ ≤ 0 there is nothing to prove. Also if r < 0 < ρ or ρ < 0 < r, then |r − ρ| ≥ |ρ| and |r − ρ| ≥ |r| so in this case, we can reduce it to two estimates with either r = 0 or ρ = 0. Also it is symmetric in r and ρ so it only remains to prove the assertion when r > ρ ≥ 0. It follows from the H¨older continuity of f (k) and the last estimate in (14.15) that for r, ρ ≥ 0,
∞
f (k) (r − 2λr)
1
−f (k) (ρ − 2λρ) (1 − 2λ)k ψ1 (λ) dλ ≤ Ck f (k) a−k,∞ |r − ρ|a−k
which proves the lemma.
(14.18)
Lemma 14.3. Let χ (t) ∈ C ∞ be a function such that χ (t) = 0, when t ≤ 0 and χ(t) = 1, when t ≥ 1, and let χθ (t) = χ (θ t). Let the Fourier transform φˆ ∈ C0∞ be 1 in a neighborhood of the origin and let χ1 ∈ C0∞ be 1 on a neighborhood of {(t, y); 0 ≤ t ≤ 2, |y| ≤ 2}. Set Sθ = χ1 φθ ∗ u, where φθ ((t, y)) = φ(θ (t, y))/θ 1+n . Then |(1 − χθ )Sθ u |a,∞ ≤ C|θ |a−b |u |b,∞ ,
(14.19)
if u is smooth and vanishes for t ≤ 0 and for t ≥ 2. Proof. First we note that by interpolation it suffices to prove the estimate for a = k an integer. Since u vanishes to infinite order as t → 0 we have if k < b ≤ k + 1, t (t − s)k−1 |u(t, y)| = (∂tk u)(s, y) ds (k − 1)! 0 t (t − s)k−1 ≤ s b−k |u |b,∞ ds ≤ Cb t b |u |b,∞ . (k − 1)! 0
(14.20)
Since φ is fast decaying we have if |α| = k, α D (1 − χθ )χ1 φθ ∗ u ≤ CN θ k−b |u |b,∞ 0
∞
|s|b ds dy (1 + |θt − s| + |θx − y|)N
(14.21)
for any N. Here the integral is uniformly bounded when θt ≤ C so the lemma follows.
378
H. Lindblad
15. The Nash Moser Iteration At this point, given the results stated in Sects. 11–14, the problem is now reduced to a completely standard application of the Nash-Moser technique. One can just follow the steps of the proof of [AG, H1, H2, K1] replacing their norms with our norms. The main difference is that we have a boundary, but we have constructed smoothing operators that satisfy the required properties for the case with a boundary. The proof of the Nash-Moser technique that we outline below is similar to the one in [L3]. The only difference is that now we also smooth in time. We will follow the formulation from [AG] which however is similar to [H1, H2]. The theorem in [AG] is stated in terms of H¨older norms, with a slightly different definition of the H¨older norms for integer values. However, the only properties that are used of the norms are the smoothing properties and the interpolation property in Sect. 14, which we proved with the usual definition, i.e. the one used in [H1]. ˜ Let us also change notation and call (u) from last section (u). For k < a ≤ k + 1, where k ≥ 0 is an integer, let |u |a,∞ = |u |H a =
|∂ α u(t, y) − ∂ α u(s, z)| + sup |u(t, y)| (15.1) |(t, y) − (s, z)|a−k (t,y)∈[0,T ]×
sup
(t,y),(s,z)∈[0,T ]× |α|=k
and |u |H 0 = sup(t,y)∈[0,T ]× |u(t, y)|. The estimates we proved for the inverse of the linearized operator and the second derivative of the operator were in terms of L∞ norms, i.e. H¨older norms for integer values. However, since |u |k,∞ ≤ |u |a,∞ ≤ |u |k+1 , if k ≤ a ≤ k + 1, it follows that they also hold for non integer values with loss of an additional derivative: (H1 ): , is twice differentiable and satisfies | (u)(v1 , v2 ) |a,∞ ≤ Ca |v1 |a+µ,∞ |v2 |µ ,∞ + |v1 |µ ,∞ |v2 |a+µ,∞ + |u |a+µ,∞ |v1 |µ ,∞ |v2 |µ ,∞ ,
(15.2)
∞ , if where µ = 3r0 + 8, for u, v1 , v2 ∈ C00
|u |µ ,∞ ≤ 1,
µ = r0 + 4,
(15.3)
where K is the constant in Lemma 12.1 ∞ satisfies (15.3) then there is a linear map ψ(u) from C ∞ to C ∞ (H2 ): If u ∈ C00 00 00 such that (u)ψ(u) = I d. It satisfies (15.4) |ψ(u)g |a,∞ ≤ Ca |g |a+λ,∞ + |g |0,∞ |u |a+λ,∞ , where λ = 2r0 + 6. Theorem 15.1. Suppose that satisfies (H1 ), (H2 ) and (0) = 0. Suppose that µ ≥ µ and let α > λ + µ + µ , α ∈ / N. Then ∞ ; |f | 2 i) There is neighborhood Wδ = {f ∈ C00 α+λ,∞ ≤ δ }, δ > 0, such that, for f ∈ Wδ , the equation (u) = f has a solution u = u(f ) ∈
∞. C00
(15.5)
Furthermore,
|u(f ) |a,∞ ≤ C |f |α+λ,∞ ,
a < α.
(15.6)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
379
∞ converging to u, that satisfies In the proof, we construct a sequence uj ∈ C00 |uj |µ ≤ 1 and |Si ui |µ ≤ 1, for all j , where Si is the smoothing operator in (15.7). The estimates (15.2) and (15.4) will only be used for convex combinations of these and hence within the domain (15.3) for which these estimates hold. Following [H1, H2, AG, K1, K2] we set
ui+1 = ui + δui , θi = θ0 2 , i
δui = ψ(Si ui )gi ,
u0 = 0,
Si = Sθi ,
θ0 ≥ 1,
(15.7)
and gi are to be defined so that ui formally converges to a solution. We have (ui+1 ) − (ui ) = (ui )(ui+1 − ui ) + ei = (ui )ψ(Si ui )gi + ei = ( (ui ) − (Si ui ))ψ(Si ui )gi + gi + ei = ei + ei + gi , (15.8) where ei = ( (ui ) − (Si ui ))δui , ei = (ui+1 ) − (ui ) − (ui )δui , ei = ei + ei .
(15.9) (15.10) (15.11)
Therefore (ui+1 ) − (ui ) = ei + gi
(15.12)
and adding, we get (ui ) =
i
gj + Si Ei + ei + (I − Si )Ei ,
j =0
Ei =
i−1
ej .
(15.13)
j =1
To ensure that (ui ) → f we must have i
gj + Si Ei = Si f.
(15.14)
j =0
Thus g0 = S0 f,
gi = (Si − Si−1 )(f − Ei−1 ) − Si ei−1
(15.15)
and (ui ) = Si f + ei + (I − Si )Ei .
(15.16)
Given u0 , u1 , . . . , ui these determine δu0 , δu1 , . . . , δui which by (15.9)–(15.10) determine e1 , . . . , ei−1 , which by (15.15) determine gi . The new term ui+1 is determined by (15.7).
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H. Lindblad
Lemma 15.2. Assume that |ui |µ ,∞ ≤ 1, |ui+1 |µ ,∞ ≤ 1, and |Si ui |µ ,∞ ≤ 1. Then |ei |r,∞ ≤ Cr |(I − Si )ui |r+µ,∞ |δui |µ ,∞ + |(I − Si )ui |µ ,∞ |δui |r+µ +Cr |Si ui |r+µ,∞ |(I − Si )ui |µ ,∞ |δui |µ ,∞ . (15.17) and
|ei |r,∞ ≤ Cr |δui |r+µ,∞ |δui |µ ,∞ + |ui |r+µ,∞ |δui |2µ ,∞ . (15.18)
Proof. The proof of (15.17) makes use of 1 ( (ui ) − (Si ui ))δui = (Si ui + s(I − Si )ui )(ui − Si ui , δui ) ds (15.19) 0
together with (15.2). The proof of (15.18) makes use of 1 (ui+1 ) − (ui ) − (ui )δui = (1 − s) (ui + sδui )(δui , δui ) ds (15.20) 0
together with (15.2).
Let α˜ > α and α˜ − µ > 2α − µ − µ . Throughout the proof Ca will stand for constants that depend on a but are independent of i and n in (15.21). Our inductive assumption (Hn ) is |δui |a,∞ ≤ δθia−α ,
0 ≤ a ≤ α, ˜
i ≤ n.
(15.21)
˜ we have |δu0 |a,∞ ≤ Cα˜ |f |α+λ,∞ ≤ If n = 0 then δu0 = ψ(0)S0 f , and if a ≤ α, ˜ . Cα˜ δ 2 , so it follows that (15.21) hold for n = 0 if we choose δ so small that Ca˜ δ ≤ θ0α−α We must now prove that (Hn ) implies (Hn+1 ) if Cα˜ δ ≤ 1, where Cα˜ is some constant that only depends on α˜ but is independent of n. Lemma 15.3. If (15.21) hold then with a constant Ca independent of i ≤ n , i
|δuj |a,∞ ≤ Ca δ min(i, 1/|α − a|) + 1)(θia−α + 1 ,
0 ≤ a ≤ α. ˜ (15.22)
j =0
Proof. Using (15.21) we get ij =0 |δuj |a,∞ ≤ Ca δ ij =0 2j (a−α) and noting that i −sj ≤ C(min (1 + 1/s, i) + 1), if s > 0, (15.22) follows. j =0 2 It follows from (15.22): Lemma 15.4. If (Hn ), i.e. (15.21), hold, and α˜ > α, then for i ≤ n + 1 we have |ui |a,∞ ≤ Ca δ(min(i, 1/|α − a|)+1)(θia−α+1), |Si ui |a,∞ ≤ Ca δ(min(i, 1/|α |(I − Si )ui |a,∞ ≤
Ca δθia−α ,
− a|) + 1)(θia−α
0 ≤ a ≤ α, ˜
where the constants are independent of n.
0 ≤ a ≤ α, ˜ (15.23)
+ 1),
a ≥ 0, (15.24) (15.25)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
381
Proof. The proof of (15.23) is just summing up the series ui+1 = ij =0 δuj , using Lemma 15.3. Equation (15.24) follows from (15.23) using (14.5) for a ≤ α˜ and (14.6), with b = α˜ for a ≥ α. ˜ Equation (15.25) follows from (14.7) with b = α˜ and (15.23) with a = α. ˜ Since we have assumed that α > µ , we note that in particular, it follows that |ui |µ ,∞ ≤ 1 and |Si ui |µ ,∞ ≤ 1, for i ≤ n + 1
if
Cµ δ ≤ 1.
(15.26)
As a consequence of Lemma 15.4 and Lemma 15.2 we get Lemma 15.5. If (Hn ) is satisfied and α > µ , then for i ≤ n, a−(2α−µ−µ )
|ei |a,∞ ≤ Ca δ 2 θi |ei |a,∞ ≤
,
0 ≤ a ≤ α˜ − µ,
(15.27)
a−(2α−µ−µ ) Ca δ 2 θi ,
0 ≤ a ≤ α˜ − µ,
(15.28)
where the constants are independent of n. As a consequence of Lemma 15.5 and (14.8) we get Lemma 15.6. If (Hn ) is satisfied, then for i ≤ n + 1, a−(2α−µ−µ )
| Si ei−1 |a,∞ ≤ Ca δ 2 θi |(Si − Si−1 )f |a,∞ ≤ |(I − Si )f |a,∞ ≤
a−β Ca θi |f |β,∞ , a−β Ca θi |f |β,∞ ,
,
a ≥ 0,
(15.29)
a ≥ 0,
(15.30)
0 ≤ a ≤ β.
(15.31)
a ≥ 0,
(15.32)
Furthermore, if α˜ − µ > (2α − µ − µ ): a−(2α−µ−µ )
|(Si − Si−1 )Ei−1 |a,∞ ≤ Ca δ 2 θi |(I − Si )Ei |a,∞ ≤
,
a−(2α−µ−µ ) Ca δ 2 θi ,
0 ≤ a ≤ α˜ − µ. (15.33)
Here the constants Ca are independent of n. Proof. Equation (15.29) follows from (15.27). For a ≤ α˜ − µ we use (14.5) with b = a and for a ≥ α˜ − µ, we use (14.6) with b = α˜ − µ. Equation (15.30) fol lows from (14.8) and (15.31) follows from (14.7). Now, Ei = i−1 j =0 ej so by Lemma ) ) α−µ−(2α−µ−µ ˜ α−µ−(2α−µ−µ ˜ i−1 ≤ Ca δ 2 j =0 θj ≤ Ca δ 2 θi , since we 15.5 |Ei |α−µ,∞ ˜ assumed that the exponent is positive. Equation (15.32) follows from this and (14.8) with b = α˜ − µ and similarly (15.33) follows from (14.7) with b = α˜ − µ. It follows that: Lemma 15.7. If (Hn ) is satisfied, α˜ −µ > (2α −µ−µ ), and α > µ then for i ≤ n+1, a−(2α−µ−µ )
|gi |a,∞ ≤ Ca δ 2 θi
Using this lemma and (15.4) we get
a−β
+ Ca θ i
|f |β,∞ ,
a ≥ 0.
(15.34)
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H. Lindblad
Lemma 15.8. If (Hn ) holds, α˜ − µ > (2α − µ − µ ), α > µ , α > λ then, for i ≤ n + 1, we have a+λ−(2α−µ−µ )
|δui |a,∞ ≤ Ca δ 2 θi
a+λ−β
+ Ca |f |β,∞ θi
,
a ≥ 0. (15.35)
Proof. Using Lemma 15.7, (15.24), and (15.4) we get |δui |a,∞ ≤ Ca |gi |a+λ,∞ + |g |λ,∞ |Si ui |a+λ,∞ a+λ−(2α−µ−µ ) a+λ−β ≤ Ca δ 2 θi + |f |β,∞ θi λ−(2α−µ−µ ) λ−β +Ca δ 2 θi δ(min(i, 1/|α − a − λ|) + |f |β,∞ θi +1)(θia+λ−α + 1).
(15.36)
Using that α > d we get (15.35).
If we now pick β = α + λ, and use the assumptions that λ + α < 2α − µ − µ , and |f |α+λ,∞ ≤ δ 2 , we get that for i ≤ n + 1, |δui |a,∞ ≤ Ca δ 2 θia−α ,
a ≥ 0.
(15.37)
If we pick δ > 0 so small that Cα˜ δ ≤ 1,
(15.38)
the assumption (Hn+1 ) is proven. The convergence of the ui is an immediate consequence of Lemma 15.3: ∞
|ui+1 − ui |a,∞ ≤ Ca δ,
(15.39)
a < α.
i=0
It follows from Lemma 15.6 that |(ui ) − f |a,∞ ≤ Ca δ 2 θia−α−λ
(15.40)
which tends to 0, as i → ∞, if a < α + λ. In particular it follows from (15.39) that |uj |µ ,∞ ≤ 1, if δ > 0 is chosen small enough. ∞ . Note that in Lemma 15.8 we proved a better estimate It remains to prove u ∈ C00 than (Hn ). In fact if we let γ = 2(α − µ) − (α + λ) > 0 and α = α + γ , then |f |α +λ,∞ ≤ C implies that
|δui |a,∞ ≤ Ca θia−α ,
a ≥ 0.
(15.41)
Using this new estimate, in place of (Hn ), we can go back to Lemma 15g.4-Lemma 15.8 and replace α by α . Then it follows from Lemma 15.8 that a+λ−2(α −µ)
|δui |a,∞ ≤ Ca θi γ
2(α
− µ) − (λ − α )
and if we now pick = use that |f |α +γ ,∞ ≤ C we see that
a+λ−β
+ Ca θ i
= 2γ and
|δui |a,∞ ≤ Ca θia−α ,
|f |β,∞ ,
α
a ≥ 0.
=
α
+γ
(15.42) = α + 2γ , and (15.43)
Since the gain γ > 0 is constant, repeating this process yields that (15.41) holds for hold for any a ≥ 0 (without δ ). It follows that any α and hence that (15.39)–(15.40) ∞ it follows uj is a Cauchy sequence in C k [0, T ] × ), for any k, and since uj ∈ C00 ∞ and (u ) → f ∈ C ∞ , and since is continuous it follows that that uj → u ∈ C00 j 00 (u) = f . Equation (15.6) follows from (15.37) with δ 2 = |f |α+λ,∞ . This concludes the proof of Theorem 15.1.
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
383
16. Existence of Initial Data Satisfying the Compatibility Conditions In this section we show that there are initial data satisfying the compatibility conditions. We do not attempt to find the most general class of initial data that do so, the purpose is simply to show that our local existence theorem is not about the empty set. The set of initial data we construct can then easily be extended to a much larger set using essentially the same proof. Let us therefore start by making some simplifying assumptions. First we assume that e(h) = h. We now want to find a formal power series solution in t of the system Dt2 xi = −∂i h,
Dt2 h − h = (∂i V j )(∂j V i ),
vi = Dt xi
(16.1)
with initial data x t=0 = f0 , Dt x t=0 = v0 , ht=0 = h0 , Dt ht=0 = h1 = − div V0 . (16.2) Lemma 16.1. Let hl = Dtl ht=0 , for l ≥ 0 and let V0 = v t=0 . If (16.1) and (16.2) holds then hl+2 = hl + Fl+2 (h−1 , . . . , hl−1 ),
l ≥ 0,
(16.3)
j
where F2 = (∂i V0 )(∂j V0i ) and in general Fl+2 = Fl+2 (h−1 , . . . , hl−1 ) is a sum of the form αn α1 n Fl+2 = Cαl11...l ...αn hl1 · · · hln , ∂xα V0ik ,
hαl = ∂xα hl ,
l ≥ 0,
(16.4)
|α1 | + l1 + . . . + |αn | + ln = l + 2, n ≥ 2, −1 ≤ li ≤ l − 1, 1 ≤ |αi | + li ≤ l + 1.
(16.5)
hα−1
=
α = (α , ik )
with
Proof. If we use that [Dt , ∂i ] = −(∂i V k )∂k we obtain from differentiating the first equation in (16.1) , ∂i Dtl h = −Dtl+1 vi + a l k1 ···kn (∂i Vk1 ) · · · (∂Vkn−1 )Vkn ,
(16.6)
where Vk = Dtk V . Here the sum is over k1 + . . . + kn = l + 2 − n, n ≥ 2, kn ≥ 1, and terms in the sum consist of contractions over n − 1 pairs of indices. From differentiating div V = ∂i V i , Dtl+1 div V = div Dtl+1 V + d l k1 ...kn (∂Vk1 ) · · · ∂Vkn ,
(16.7)
where the sums are over k1 + . . . + kn = l + 2 − n, n ≥ 2 and terms in the sum consist of contractions over n pairs of indices. It follows from (16.1) that Dt (Dt h + div V ) = 0. Hence Dtl h−Dtl+2 h = e l k1 ...kn (∂Vk1 ) · · · ∂Vkn +d l k1 ...kn (∂Vk1 ) · · · (∂ 2 Vkn−1 )Vkn , (16.8) where kn ≥ 1 in the last sum. Finally we note that we can turn Vk+1 , for k ≥ 1 into hk :
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H. Lindblad
Vk+1 = −∂hk + a kk1 ...kn (∂Vk1 ) · · · (∂Vkn−1 )Vkn ,
(16.9)
where kn ≥ 1 and V1 = −∂h. This proves the general form (16.4) and it remains to prove the range of the indices in (16.5). To prove this we note that the terms in (16.8) are contractions over n pairs of indices and this is still true for the terms we obtain by using (16.9) to replace the factors of V by factors of h. This proves that |α1 | + . . . + |αn | = 2n. On the other hand when we replace factors of Vk+1 by hk the number of time derivatives goes down by one for each factor, so we conclude that l1 + . . . + ln = l + 2 − 2n. This proves (16.5) apart from the last statement. That |αi | ≥ 1 is clear and if li = −1 then |αi | ≥ 2, in view of (16.4), so in general |αi | + li ≥ 1, and since n ≥ 2, (16.5) follows. We will now obtain a formal power series solution in the distance to the boundary of the system for hl in Lemma 16.1. In order to do this, we will first choose simpler initial data for (16.2), f0 = y,
v0 = ∂φ,
k0 = − φ,
(16.10)
where φ is to be determined. Let h−1 = −φ, then (16.3) hold also for l = −1 with F1 = 0. Lemma 16.2. Suppose that gab = δab . Suppose also that h0, l and h1, l are smooth for l ≥ −1, and let Fl be as in Lemma 16.1, and F1 = 0. Then the system hl = hl+2 + Fl+2 (h−1 , . . . , hl−1 ), with boundary conditions hl = h0, l , ∂
∇N hl ∂ = h1,l
l ≥ −1
(16.11)
l ≥ −1
(16.12)
has a formal power series solution in the distance to the boundary: hl (r, ω) ∼ hn, l (ω)(1 − r)n /n!.
(16.13)
Let χ be smooth such that χ (d) = 1, when |d| ≤ 1, χ (d) = 0, when |d| ≥ 2 and χ ≥ 0. Then there are εln > 0 such that hl (r, ω) = χ ((1 − r)/εln )hn, l (ω)(1 − r)n /n! (16.14) are smooth functions and such that (16.11) hold to infinite order at the boundary. Proof. We have = ∇N2 + tr θ ∇N + ,
(16.15)
where θ is the second fundamental form of the boundary and is the tangential Laplacian on the boundary. In the case gab = δab , ∇N = ∂r and tr θ = (n − 1)/r, where r is the radial derivative. Furthermore = r −2 ω , where ω is the angular Laplacian on S n−1 . Hence we have the system ∂r2 hl = − r12 ω hl + hl+2 −
n−1 r ∂r hl
+ Fl+2 (h−1 , . . . , hl−1 ),
We want this system to be satisfied to all orders at the boundary, so if Fm,l = ∂ m Fl (h−1 , . . . , hl−3 ) hk, l = ∂ k hl , r
∂
r
∂
l ≥ −1. (16.16)
(16.17)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
we want hk, l =
cij km iω hm, l+2j +
2i+2j +m≤k, m≤k−1
385
dij km iω Fm, l+2j (16.18)
2i+2j +m≤k
to hold for all k ≥ 2 and l ≥ −1, where h0, l and h1, l are the given boundary conditions. We now want to use induction. Note that the first term in the right of (16.18) contains hk ,l for k +l ≤ k +l, and k < k, and the second term contains hk ,l for k +l < k +l, by the last inequality in (16.5). Assume that we found hk, l ,
for
k + l ≤ N,
l ≥ −1,
k≥0
(16.19)
such that (16.18) holds for k + l ≤ N and k ≥ 2. Note that if N = 0 then k ≤ 1 so there is nothing to prove. We now want to find hk, l for k + l = N + 1 such that (16.18) hold also for k + l ≤ N + 1. This is again proven by induction. Assume that in addition to (16.19) we found hk, l ,
for
k + l = N + 1,
and
0 ≤ k ≤ M,
l ≥ −1
(16.20)
such that (16.18) hold for k ≥ 2. Note that for M ≤ 1 there is nothing to prove. Since m in the first sum on the right of (16.18) is less than k in the left it follows that we can find hk+1, l−1 such that (16.18) hold. In order to prove (16.14) we note that hm,l are smooth functions on S n−1 . Hence we can use the usual trick of choosing εml so small that ( hm,l m+1 + 1) εml ≤ 1/2, in which case the sum converges in H m for any m and r. Now, we want to find a formal power series solution in t of the system (16.1) with initial data in (16.2) of the form f0 (y) = f˜0 (y) + y, v0 = v˜0 − ∂ h−1 , ˜ h1 = − div V0 + h−1 ,
h0 = h˜ 0 + h0 , (16.21)
where h0 , h−1 are given by Lemma 16.2 and f˜0 , v˜0 and h˜ 0 vanish to infinite order at the boundary. Let hl , for l ≥ 2 be defined by (16.3). Then it inductively follows that hl = h˜ l + hl ,
(16.22)
where hl are as in Lemma 16.2 and h˜ l vanish to infinite order at the boundary. Therefore if we choose boundary data in (16.12) such that h0, l = 0 for l ≥ 0,h1,0 ≤ c0 < 0. Then it follows that we can choose h˜ 0 so that h0 > 0 in and ∇N h0 ∂ ≤ −c0 < 0. Moreover it follows that hl ∂ = 0, l ≥ 0, (16.23) and hence the compatibility conditions are satisfied to all orders. We can now construct smooth functions in [0, T ]×, satisfying the initial conditions (16.2) and Eqs. (16.2) to infinite order as t → 0: Lemma 16.3. Suppose that initial data f0 , v0 and h0 and h1 = − div V0 satisfy the compatibility conditions for all orders, i.e. if hl for l ≥ 2 are defined by (16.3) then hl ∂ = 0, for l ≥ 0. (16.24) Then there are smooth functions (x, h) in [0, T ] × , such that (16.2) hold, (16.1) is satisfied to infinite order as t → 0 and h∂ = 0.
386
H. Lindblad
Proof. Let χ be as in Lemma 16.2 and set h(t, y) =
∞
χ (t/εl )hl (y)t l / l!,
(16.25)
l=0
it follows that where εl > 0 are chosen so that ( hl m + 1)εm ≤ 1/2. Then the sum converges in H m for any m so h is smooth and satisfies h∂ = 0 and Dtl ht=0 = hl . Furthermore let x(t, y) be defined by Dt2 xi = −∂i h, xi t=0 = f0 , Dt x t=0 = v0 . (16.26) 17. The General Case when the Enthalpy is a Strictly Increasing Function of the Density We will now outline how to generalize the existence result obtained for e(h) = ch to the case when e(h) is a smooth strictly increasing function satisfying 1/c1 ≤ e (h) ≤ c1 . First we will show that the functional h = (x), i.e. the solution of (2.21), exist for x in a bounded set, |u |r0 +r1 ,∞ ≤ 1, and for T sufficiently small. Since in the Nash-Moser iteration we need a bound for |h |3,∞ , we will first show that we can obtain such a bound as well as a bound for h r0 +3 independent of T . Local existence for the nonlinear wave equation follows from a standard argument using essentially the same estimates, so it is just a question of showing that we have a priori bounds up to some time T > 0 that only depend on the approximate solution (x0 , h0 ) and is independent of x = u + x0 as long as |u |r0 +r1 ,∞ ≤ 1. Once we have a bound for |h |4,∞ , the bound for higher derivatives follows from this. The equation we study in this section is ht=0 = 0, f = (∂i V j )(∂j V i ), (17.1) Dt e (h)Dt h − h = f, where e + 1/e ≤ c1 ,
|g ab | + |gab | ≤ c1 ,
|∂x/∂y|2 + |∂y/∂x|2 ≤ c1 (17.2)
a,b denote a continuous function of c that also for some constant 0 < c1 < ∞. Let K10 1 depends on the order of differentiation r but is independent of a lower bound for T . We prove the following theorem:
Theorem 17.1. Let r0 = [n/2] + 1 be the Sobolev exponent and let k ≥ 1. There are continuous function Ck and Dk such that if T > 0 is so small that T C1 ( |x0 |r0 +2+1,∞ , |h0 |r0 +2+1,∞ ) ≤ 1,
(17.3)
and u = x − x0 is so small that |u |r0 +2+k,∞ ≤ 1,
(17.4)
then (17.1) has a smooth solution for 0 ≤ t ≤ T satisfying |h |k,∞ ≤ Dk ( |x0 |r0 +2+k,∞ , |h0 |r0 +2+k,∞ ).
(17.5)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
387
Furthermore, there is a continuous function K2 of |x |2,∞ + |h |2,∞ + |h0 |2,∞ + 1/T + c1
(17.6)
depending also on r such that |h |r,∞ ≤ K2 |x |r+r0 +2,∞ + |h0 |r+r0 +2,∞ .
(17.7)
Proof. Since by Sobolev’s Lemma |h |1,∞ ≤ C( h 1,r0 + h 0,r0 +1 ),
(17.8)
where C is independent of T , It follows from Lemma 17.2 that d E˜ r /dt ≤ Mr (E˜ r+1 + 1)r+2 ,
r ≥ r0 ,
(17.9)
(r+2)2
|x | −(r+1) /dt ≤ ˜ where Mr = K10 r+3,∞ . Integrating this inequality gives −d(Er+1 +1) Mr (r + 1) and hence (E˜ r+1 (t) + 1)−(r+1) ≥ (E˜ r+1 (0) + 1)−(r+1) − Mr (r + 1)t. If −(r+1) ˜ ˜ Mr (r + 1)t ≤ (E˜ r+1 (0) r+ 1)
/2 it follows that Er+1 (t) + 1 ≤ 2(Er+1 (0) + 1). ˜ Since Er+1 (0) ≤ K20 s=0 ∂x r−s,∞ ( h0 1,s + h0 0,s+1 ) , this proves the first part of the theorem for k = 1 and for k ≥ 2 it follows from also using Theorem 17.3. It follows from Lemma 17.2 and interpolation in space time that
d Eˆ r+1 ≤ K2 Eˆ r+1 + |x |r+3,∞ , dt
r if Eˆ r+1= |x |r+2−s E˜ s+1 . (17.10) s=0
Multiplying by the integrating factor eK2 t and integrating, we get Eˆ r+1 (t) ≤ K2 Eˆ r+1 (0) + |x |r+3,∞ .
(17.11)
By Theorem 7.3 h r+1 ≤ K2 Eˆ r+1 and hence h(t, ·) r+1 ≤ K2
r
|x |r+2−s h0 (0, ·) s+1 + |x |r+3 .
(17.12)
s=0
Using Sobolev’s lemma and interpolation the estimate for |h |r,∞ follows.
Lemma 17.2. Suppose that g ab and e = e (h) are smooth and satisfy (7.4). For s ≥ 0 let 1 1/2 Es+1 (t) = e (Dts+1 h)2 + gab (Dˆ ts H a )(Dˆ ts H b )κdy , 2 H a = g ab ∂b h, E0 (t) =
h
2 κdy
and E˜ r+1 =
h 1,r + h 0,r+1 E˜ 1+r
(17.13) r+1 s=0
Es . Then for r ≥ 0 ,
≤ C rs=0 ∂x r−s,∞ E˜ 1+s ,
≤ C rs=0 ∂x r−s,∞ h 1,s + h 0,s+1 ,
(17.14) (17.15)
388
H. Lindblad
and r+1 min(r+1−k,r)
d E˜ r+1 ≤ K10 h k1,∞ ∂x r+1−k−s,∞ E˜ s+1 dt s=0 k=0
+ ∂x r+2−k,∞ .
(17.16)
Proof. The proof is similar to that of Lemma 7.2. We have 2 dEs+1 e (Dts+1 h)(Dts+2 h) + gab (Dˆ ts H a )(Dˆ ts+1 H b ) κ dy = dt 1 ˆ + (Dt e )(Dts+1 h)2+(Dˇ t gab )(Dˆ ts H a )(Dˆ ts H b ) κ dy. (17.17) 2 ( h 2 Here the terms on the second row are bounded by K10 1,∞ + g 1,∞ )Es+1 . The first row of the right-hand side is up to lower order (Dts+1 h) Dˆ ts Dt (e Dt h) + (Dˆ ts H a )(∂a Dts+1 h) κ dy = (Dts+1 h) Dˆ ts Dt (e Dt h) − κ −1 ∂a κ Dˆ ts H a κdy = (Dts+1 h)(Dˆ ts f )κdy, (17.18)
where we have integrated by parts using that Dts+1 h∂ = 0, that Dt e Dt h − κ −1 ∂a a κH = f and that Dˆ ts κ −1 ∂a κH a = κ −1 ∂a κ Dˆ ts H a . In fact, using Lemma 6.3 we get, since H a = ∂a h, Dˆ ts+1 H a − g ab ∂a Dts+1 h s s + 1 ab ˇ s+1−i =− g (Dt gbc )Dˆ ti H c i
(17.19)
i=0
and, since Dˆ t Dt2 e(h) = κ −1 Dts κDt2 e(h)) , Dˆ ts Dt2 e(h) − e (h)Dts+2 h s−1 s −1 s−i = κ (Dt κ)(Dt2+i e(h)) + Dt2+s e(h) − e (h)Dts+2 h. i
(17.20)
i=0
Here the right-hand side is bounded by a constant times s−1 i=0
κ −1 |Dts−i κ|
|e(k) (h)| |Dts1 h|· · · |Dtsk h|
s1 +...+sk =i+2, si ≥1, k≥1
+
|e(k) (h)| |Dts1 h|· · · |Dtsk h|.
s1 +...+sk =s+2, si ≥1, k≥2
(17.21)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
389
The L2 of this can be estimated by Theorem 7.3: K1,0
s+1
min(s+2−k,s+1)
h k1,∞
k=0
+
∂x
i=1
∂x
s+2−k−i,∞
h 0,i + h 1,i−1
(17.22)
.
s+2−k,∞
s ab s−i ˇ ˆ i cb By Lemma 6.3 Dˆ ts (g ab ∂b h) = g ab ∂b Dts h + s−1 so it i=0 i g (Dt gbc )Dt (g
∂b h)
s s s follows that ∂Dt h ≤ i=0 ∂x r−i E˜ 1+i Since also Dˆ t f ≤ K10 ∂x s+2 the lemma follows. Theorem 17.3. Suppose that Dt2 e(h) − h = f , where f = (∂i V j )(∂j V i ). Suppose also that |e(k) (h)| ≤ Ck . Then we have r−k r
j
h r ≤ K1,0 h k1,∞ ∂x r−k−i,∞ h i−1,1 + h i,0 k=j −1
+
∂x
i=1
(17.23)
r−k,∞
and r−k r
j
h r ≤ K1,0 ∂x r−k−i,∞ h 0,i + h 1,i−1 h k1,∞ k=j −1
+
∂x
r−k,∞
where j
[h]r =
∂x
r,∞
i=1
(17.24)
,
|h|r1 · · · |h|rk ,
r1 +...+rk =r, ri ≥1, k≥j
=
|∂x |r1 ,∞ · · · |∂x |rk ,∞ .
(17.25)
r1 +...+rk =r, ri ≥1
Proof. The first inequality follows from Lemma 17.4 below and interpolation and the second follows from the first and Lemma 17.6 below. Lemma 17.4. Suppose that Dt2 e(h) − h = f , where f = (∂i V j )(∂j V i ). Suppose also that |e(k) (h)| ≤ Ck . With notation as in Definition 5.1 we have
(17.26) ∂x r+s−i−j h i,(j,1) , [h]r,s ≤ K10
where ∂x
l
i+j ≤r+s
is as in Definition 5.1,
h r,s =
|h|r1 ,s1 · · · |h|rk ,sk ,
r1 +...+rk =r, si +...+sk =s si +ri ≥1
h r,(s,m) =
|h|r1 ,s1 · · · |h|rk ,sk ,
r1 +...+rk =r, si +...+sk =s si +ri ≥1, si ≤m,
(17.27)
390
H. Lindblad
and h 0,0 = 1, h 0,(0,1) = 1. Proof. If h = η(e) is the inverse of e(h), then |h|r,s ≤ C
|η(k) (e)| |e|r1 ,s1 · · · |e|rk ,sk .
(17.28)
r1 +...+rk =r, s1 +...+sk =s, ri +si ≥1
Since, |e|r,s ≤ | h|r,s−2 + |f |r,s−2 , where
| h|r,s−2 ≤ K10
∂x
1≤i≤r+2, j ≤s−2
r+s−i−j
|h|i,j ,
s ≥ 2,
(17.29)
s ≥ 2,
(17.30)
and
|f |r,s−2 ≤ K10 ∂x r+s , we obtain
|e|r,s ≤ K10
∂x
i≤r+2, j ≤s−2
It follows that
h r,(s,m) ≤ K10
∂x
r+s−i−j
i+j ≤r+s
r+s−i−j
[h]i,j ,
s ≥ 2,
(17.31)
s ≥ 2.
(17.32)
m ≥ 2, (17.33)
[h]i,(j,m−1) ,
and the lemma follows by induction.
Lemma 17.5. For r ≥ 1 we have with notation as in Definition 5.3, r
h r ≤ K10
j
h 1,∞
j =0
+
∂x
r−j
∂x
r−i−j,∞
h 0,i + h 1,i−1
i=1
r−j,∞
(17.33)
.
Proof. We have | h|r,s ≤ |e(h)|r,s+2 + |f |r,s ,
(17.34)
where |e(h)|r,s ≤ e (h)|h|r,s + C k = [h]r,s
r+s
k |e(k) (h)| [h]r,s ,
where
k=2
|h|r1 ,s1 · · · |h|rk ,sk ,
r1 +...+rk =r, s1 +...+sk =s ri +si ≥1
(17.35)
Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary
391
and by the previous lemma and interpolation in space only
j k ≤ K10 ∂x r+s−i−j,∞ h 1,∞ h i,0 [h]r,s i+j ≤r+s, j ≥k−1,i≥1
+ h i−1,1 .
(17.36)
Using Theorem 6.1 we get for r ≥ 2
∂x r+s−i−j,∞ h i,j +2 h r+2,s ≤ K10 i≤r, j ≤s
+K10
s
∂x
j =0
+K10
∂x
r+s+1−j,∞
h 1,j,∞
+K10
r+s+2,∞
∂x
i+j ≤r+s+2, i,j ≥1
r+s+2−i−j,∞
j h 1,∞ h i,0
+ h i−1,1 .
(17.37)
Hence using induction, we get
∂x r−i,∞ h 0,i + h 1,i−1 h r ≤ K10 1≤i≤r +K10 +K10
∂x r,∞
∂x
i+j ≤r, i,j ≥1
j
r−i−j,∞
h 1,∞ h i .
Since h 1 ≤ h 0,1 + h 1,0 the lemma now follows from induction.
(17.38)
Alternatively, one can use interpolation in space-time. Lemma 17.6. We have 1−l/m
l/m
|Dtl h |L2/(m−l) ≤ CT |h |L∞ |Dtm h |L2
m | |Dtl1 h| · · · |Dtlk h| |L2 ≤ CT |h |k−1 L∞ |Dt h |L2 ,
,
(17.39) (17.40)
where m = l1 + . . . + lk . Furthermore, if h = h˜ + h0 , where h˜ vanishes to infinity order as t → 0 then, with a constant independent of T but depending on h0 we have m
| |Dtl1 h| · · ·|Dtlk h| |L2 ≤ Cm ( |h |∞ , |h0 |∞,m )
j
|Dt h |L2 + 1). (17.41)
j =0
Proof. The interpolation inequalities are standard and it is also standard that the constant is independent of T if h vanishes to infinite order as t → 0. Hence in the product we ˜ 2/(m−li ) + |Dtli h0 |L∞ . can estimate |Dtli h |L2/(m−li ) ≤ |Dtli h | L Acknowledgements. I would like to thank Demetrios Christodoulou, Richard Hamilton and Kate Okikiolu for many long and helpful discussions.
392
H. Lindblad
References [AG] Alinhac, S., Gerard, P.: Operateurs pseudo-differentiels et theorem de Nash-Msoer. Paris: Inter Editions and CNRS, 1991 [BG] Baouendi, M.S., Gouaouic, C.: Remarks on the abstract form of nonlinear Cauchy-Kovalevsky theorems. Comm. Part. Diff. Eq. 2, 1151–1162 (1977) [C1] Christodoulou, D.: Self-Gravitating Relativistic Fluids: A Two-Phase Model. Arch. Rat. Mech. Anal. 130, 343–400 (1995) [C2] Christodoulou, D.: Oral Communication, August, 1995 [CK] Christodoulou, D., Klainerman, S.: The Nonlinear Stability of the Minkowski space-time. Princeton, NJ: Princeton Univ. Press, 1993 [CL] Christodoulou, D., Lindblad, H.: On the motion of the free surface of a liquid. Comm. Pure Appl. Math. 53 1536–1602 (2000) [CF] Courant, R., Friedrichs, K.O.: Supersonic flow and shock waves. Berlin-Heidelberg-NewYork: Springer-Verlag, 1977 [Cr] Craig, W.: An existence theory for water waves and the Boussinesq and Korteweg-deVries scaling limits. Comm. in P. D. E. 10, 787–1003 (1985) [DM] Dacorogna, B., Moser, J.: On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincare Anal. Non. Lineaire 7, 1–26 (1990) [DN] Dain, S., Nagy, G.: Initial data for fluid bodies in general relativity. Phys. Rev. D 65(8), 084020 (2002) [E1] Ebin, D.: The equations of motion of a perfect fluid with free boundary are not well posed. Comm. Part. Diff. Eq. 10, 1175–1201 (1987) [E2] Ebin, D.: Oral communication, November, 1997 [F] Friedrich, H.: Evolution equations for gravitating ideal fluid bodies in general relativity. Phys. Rev. D 57, 2317 (1998) [FN] Friedrich, H., Nagy, G.: The initial boundary value problem for Einstein’s vacuum field equation. Commmun. Math. Phys. 201, 619–655 (1999) [Ha] Hamilton, R.: Nash-Moser Inverse Function Theorem. Bull. Amer. Math. Soc. (N.S.) 7, 65–222 (1982) [H1] H¨ormander, L.: The boundary problem of Physical geodesy. Arch. Rat. Mech. Anal. 62, 1–52 (1976) [H2] H¨ormander, L. : Implicit function theorems. Lecture Notes, Stanford Univ, 1977 [H3] H¨ormander, L.: The analysis of Linear Partial Differential Operators III. Berlin-Heidelberg-NewYork: Springer Verlag, 1985 [K1] Klainerman, S.: On the Nash-Moser-H¨ormander scheme. Unpublished lecture notes [K2] Klainerman, S.: Global solutions of nonlinar wave equations. Comm. Pure Appl. Math. 33, 43– 101 (1980) [L1] Lindblad, H.: Well posedness for the linearized motion of an incompressible liquid with free surface boundary. Comm. Pure Appl. Math. 56, 153–197 (2003) [L2] Lindblad, H.: Well posedness for the linearized motion of a compressible liquid with free surface boundary. Commun. Math. Phys. 236, 281–310 (2003) [L3] Lindblad, H.: Well posedness for the motion of an incompressible liquid with free surface boundary. To appear in Annals of Math., July 2005 [Na] Nalimov, V.I.: The Cauchy-Poisson Problem (in Russian). Dynamika Splosh. Sredy 18, 104–210 (1974) [Ni] Nishida, T.: A note on a theorem of Nirenberg. J. Diff. Geom. 12, 629–633 (1977) [R] Rendall, A.D.: The initial value problem for a class of general relativistic fluid bodies. J. Math. Phys. 33, 1047–1053 (1992) [SY] Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. Cambridge, MA: International Press, 1994 [S] Stein, E.: Singular Integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press, 1970 [W1] Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39–72 (1997) [W2] Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc. 12, 445–495 (1999) [Y] Yosihara, H.: Gravity Waves on the Free Surface of an Incompressible Perfect Fluid. Publ. RIMS Kyoto Univ. 18, 49–96 (1982) Communicated by P. Constantin
Commun. Math. Phys. 260, 393–401 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1411-9
Communications in
Mathematical Physics
The Lorenz Attractor is Mixing Stefano Luzzatto1 , Ian Melbourne2 , Frederic Paccaut3 1 2 3
Department of Mathematics, Imperial College London, SW7 2AZ, U.K. Department of Mathematics and Statistics, University of Surrey, Guildford GU2 5XH, U.K. LAMFA CNRS UMR 6140, University of Picardie, 33, rue Saint Leu, 80039 Amiens, France
Received: 8 October 2004 / Accepted: 25 March 2005 Published online: 9 August 2005 – © Springer-Verlag 2005
Abstract: We study a class of geometric Lorenz flows, introduced independently by Afra˘ımoviˇc, Bykov & Sil nikov and by Guckenheimer & Williams, and give a verifiable condition for such flows to be mixing. As a consequence, we show that the classical Lorenz attractor is mixing.
1. Introduction and Statement of Results 1.1. The Lorenz equations. Many systems of nonlinear differential equations that were first studied almost 50 years ago and which were motivated mainly by problems in geophysical and astrophysical fluid dynamics and dynamical meteorology [2, 8, 15, 23], remain difficult to understand rigorously to the present day. In 1963, Lorenz [17] introduced the following system of differential equations: x˙ = 10(y − x), y˙ = 28x − y − xz, z˙ = xy − 83 z.
(1.1)
Approximate numerical studies of these equations led Lorenz to emphasise the possibility and importance of sensitive dependence on initial conditions even in such simplified models of natural phenomena. A combination of results obtained over the last 25 years [1, 14, 20, 25, 30] and culminating in the work of Tucker [27, 28] gives the following statement (see [26, 29] for detailed surveys): The Lorenz equations admit a robust attractor A which supports a “physical” ergodic invariant probability measure ν with a positive Lyapunov exponent.
394
S. Luzzatto, I. Melbourne, F. Paccaut
Recall that the measure ν is called physical, or Sinai-Bowen-Ruelle (SRB), if for Lebesgue almost every solution u(t) ∈ R3 starting close to A and all continuous functions h : R3 → R, 1 T →∞ T
lim
T
h(u(t)) dt =
0
h dν. A
In this paper we take a further step in the understanding of the statistical properties of the Lorenz attractor. A measure ν is mixing for a flow t if ν(t (A) ∩ B) → ν(A)ν(B) for all measurable sets A, B, as t → ∞. We say that the Lorenz attractor is mixing if the SRB measure ν mentioned above is mixing. We prove the following Theorem 1. The Lorenz attractor is mixing. In fact, our result shows that the Lorenz attractor is stably mixing: sufficiently small C 1 perturbations of the flow are mixing. There are relatively few explicit examples of flows that have been proved to be mixing. Anosov [4] showed that geodesic flows for compact manifolds of negative curvature are mixing, and this was generalised [16] to include contact flows. Moreover, mixing persists under C 1 perturbations. For codimension one Anosov flows [21] and for Anosov flows with a global infranil cross-section [5], the set of mixing flows is C 1 open, but the corresponding result for general Anosov flows is not known. Recently [11] it was shown that mixing holds for a C 2 -open and C r -dense set of C r uniformly hyperbolic (Axiom A) flows for all r ≥ 2. (In these references, mixing is proved for any equilibrium measure for a H¨older potential.) However, the conditions for stable mixing in [11] are not easily verifiable. The Lorenz attractor is an example of a singular hyperbolic attractor [18] (uniformly hyperbolic, except for a singularity due to the attractor containing an equilibrium). Somewhat surprisingly, we show that the singular nature of the Lorenz attractor assists in the search for a verifiable condition for mixing. 1.2. Geometric Lorenz attractors. Afra˘ımoviˇc, Bykov and Sil nikov [1] and Guckenheimer and Williams [14, 30] introduced a geometric model that is an abstraction of the numerically-observed features possessed by solutions to (1.1). Tucker [27, 28] proved that the geometric model is valid, so the Lorenz equations define a geometric Lorenz flow. Accordingly, our approach to Theorem 1 is to establish mixing for geometric Lorenz flows satisfying certain hypotheses, and then to verify from [27, 28] that the hypotheses are satisfied for the Lorenz equations. Roughly speaking, a geometric Lorenz flow is the natural extension of a geometric Lorenz semiflow which is itself a suspension flow built over a certain type of one-dimensional expanding map f . Moreover, the roof function r has a logarithmic singularity (due to the equilibrium for the flow). Precise definitions are given in Sect. 2. In the literature, a standard assumption is that the map f is locally eventually onto (l.e.o.), see Definition 3. We prove that this is a sufficient condition for the corresponding flow to be mixing.
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Theorem 2. Let t be a geometric Lorenz flow. Suppose that the associated one-dimensional map f is l.e.o. Then t is mixing (and even Bernoulli). Remark 1.1. It was shown in [19] that geometric Lorenz flows are generically mixing (albeit in the C 1 topology). Stability of mixing is not proved in [19], nor is it shown there that the actual Lorenz attractor is mixing. 1.3. Outline of the proof of Theorem 2. The geometric Lorenz flow t possesses a strong stable foliation and quotienting by the foliation yields a geometric Lorenz semiflow ft . It suffices to prove that the semiflow ft is mixing. Moreover, weak mixing of ft implies that the original flow is mixing (in fact Bernoulli) by a general result of Ratner [22]. As mentioned above, ft is a suspension semiflow over a one-dimensional map f , where the roof function r possesses a logarithmic singularity. Label the singular point 0. Using spectral characterisations of weak mixing, it suffices to show that the cohomological equation ψ ◦ f = eiar ψ has no measurable solutions for all a > 0. A recent Livˇsic regularity theorem by Bruin et al. [6] shows that if such an eigenfunction ψ exists, then it must be continuous. This, coupled with the regularity of f and r away from 0 and the singularity of r at 0, is used to obtain a contradiction. The remainder of the paper is structured as follows. In Sect. 2, we discuss the geometric Lorenz flow and its relation to the Lorenz attractor. In particular, we indicate how Theorem 1 follows from Theorem 2. In Sect. 3, we prove Theorem 2. In Sect. 4, we discuss extensions of our main results and some related future directions. 2. The Lorenz Attractor is a Geometric Lorenz Flow In this section we collect and organise several results from the existing literature on the relation between the Lorenz attractor and geometric Lorenz flows. We recall first of all some basic relevant facts about the Lorenz equations (1.1), see [26]. The origin is an equilibrium of saddle type with two negative (stable) and one positive (unstable) eigenvalues λss < λs < 0 < λu . It is also the case that λu > |λs |. Suppose that a finite number of nonresonance conditions are satisfied so that the vector field is smoothly linearisable in a neighbourhood of 0. In these coordinates, the flow near 0 has the form (x1 , x2 , x3 ) → (eλu t x1 , eλss t x2 , eλs t x3 ). By a linear rescaling, we can suppose that the domain of linearisation of the flow includes the cube [−1, 1]3 . = {(x1 , x2 , x3 ) : x1 = Define = {(x1 , x2 , x3 ) : |x1 |, |x2 | ≤ 1, x3 = 1} and by ±1, |x2 |, |x3 | ≤ 1}. Then we define the first hit map P0 : → x2 , x3 ), P0 (x1 , x2 , 1) = (eλu t0 x1 , eλss t0 x2 , eλs t0 ) = (sgn x1 , . Solving eλu t0 x1 = sgn x1 for t0 = where t0 is the “time of flight” from to −(ln |x1 |)/λu , we obtain P0 (x1 , x2 , 1) = (sgn x1 , |x1 |β x2 , |x1 |α ), is where α = |λs |/λu ∈ (0, 1) and β = |λss |/λu > 0. Note that P0 : → well-defined on \ W s (0), where W s (0) = {(0, x2 , 1) : |x2 | ≤ 1}, and that t0 has a logarithmic singularity at x1 = 0. Tucker [27, 28] proves that there exists a compact trapping region N ⊂ such that the Poincar´e first return map P : N \ W s (0) → N is well defined. We can decompose
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and . Note that P1 is a diffeoP = P1 ◦ P0 , where P1 is the first-hit map between morphism where it is defined, and that the time of flight t1 for P1 is bounded. Hence the time of flight t = t0 + t1 for the full return map P is smooth except for a logarithmic singularity at x1 = 0. Moreover, the following crucial hyperbolicity estimate holds. Lemma 2.1 ([28]). The return map P admits a forward invariant cone field. In other words, there exists a cone C(u) inside at each point u ∈ N \W s (0) such that (dP )u C(u) is strictly contained inside C(P u). Moreover, there exist constants c > 0, τ > 1, such that for each u ∈ N \ W s (0), (dP n )u v ≥ cτ n v , for every v ∈ C(u) and n ≥ 1. A consequence of Lemma 2.1 is that the return map P has an invariant C 1+ε stable foliation for some ε > 0. Let I = [−1, 1]. We obtain a singular one-dimensional map f : I → I by quotienting along stable leaves. Let J = I \ {0}. Then it is immediate that on J : f(i) f is C 1+ε , f(ii) |f (n) | ≥ cτ n for all n ≥ 1, f(iii) C −1 |x|α−1 ≤ f (x) ≤ C|x|α−1 , where c > 0, τ > 1 are the constants in Lemma 2.1, α = |λs |/λu ∈ (0, 1), and C ≥ 1 is a constant. Remark 2.2. An explicit construction of f can be obtained as follows. Let I ⊂ N denote a global cross-section to the stable foliation with 0 ∈ I . Coordinates on I can be chosen so that I = [−1, 1]. For each x ∈ I , let W s (x) ⊂ N denote the corresponding stable leaf. Then P (W s (x)) ⊂ W s (P x). Define f x to be the unique intersection point of W s (P x) with I . The underlying flow t possesses an invariant strong stable foliation corresponding to the stable foliation for P . Quotienting the flow along strong stable leaves yields a semiflow ft with (noninvertible) Poincar´e map f : I → I as above and return time function r : I → R+ with a logarithmic singularity at 0. More precisely, r satisfies r(i) r(x) → ∞ as x → 0, r(ii) |r(x) − r(y)| ≤ C| ln |x| − ln |y|| for all x, y > 0 and all x, y < 0. Note that it follows from r(ii) that r is Lebesgue integrable and r is C 1 on J . We have described how to pass from the original three-dimensional flow defined by the Lorenz equations (1.1) to a two-dimensional Poincar´e map P : → and then to a one-dimensional expanding map f : I → I . The process is reversible: taking the natural extension of f : I → I recovers the Poincar´e map P : N → N and taking the suspension of the P by the roof function t : N → R+ recovers the original Lorenz flow. Alternatively, these steps can be carried out in the opposite order to recover the Lorenz flow as the natural extension of the suspension semiflow of the map f : I → R by the roof function r : I → R+ . The latter viewpoint is the geometric Lorenz flow construction [14, 30]. For completeness, the notions of suspension and natural extension are recalled at the end of this section. Definition 1. Let f : I → I . Assume that f (0) is undefined, with f (0+ ) = −1 and f (0− ) = +1. It is assumed that f (1) ∈ (0, 1) and f (−1) ∈ (−1, 0). (See Fig. 1.) If conditions f(i)–f(iii) are satisfied, then f is a Lorenz-like expanding map.
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Fig. 1. The graph of a Lorenz-like expanding map f : [−1, 1] → [−1, 1]
Definition 2. A semiflow ft is called a geometric Lorenz semiflow if it is the suspension over a Lorenz-like expanding map f : I → I by a roof function r : I → R+ satisfying conditions r(i), r(ii). A flow t is called a geometric Lorenz flow if it is the natural extension of a geometric Lorenz semiflow. The results of Tucker described above imply that the Lorenz equations indeed define a geometric Lorenz flow. Moreover, as outlined below, Tucker showed that the associated expanding map f : I → I is transitive for the Lorenz equations, thus establishing the existence of the Lorenz attractor. In the following definition, we stress that f (0) is undefined. In particular, the endpoints ±1 have no preimages. Definition 3. The map f : I → I is locally eventually onto (l.e.o.) if for any open set U ⊂ J , there exists k ≥ 0 such that f k U contains (0, 1). Remark 2.3. Since f (0, 1) contains (−1, 0) and vice versa, we could equally use (−1, 0) instead of (0, 1) in the definition of l.e.o. Clearly, the l.e.o. property implies that f is topologically transitive. The exact formulation of l.e.o. varies in the literature, but our definition agrees with the one in [12]. √ Remark 2.4. It was anticipated in [30] that f(ii) would hold with c = 1, τ > 2. It is easy to check that this is a sufficient condition for f to be l.e.o. Surprisingly [28], for the actual Lorenz equations it turns out that f (x) < 1 for certain values of x ∈ I . Nevertheless, it is still the case that f is l.e.o., see Tucker [28]. It can be shown that Lorenz-like expanding maps satisfying the l.e.o. condition have a unique ergodic probability measure µ that is equivalent to Lebesgue (see for example Sect. 3). The suspended measure ν = µr defines an SRB measure for the geometric Lorenz flow. In particular, Theorem 1 follows from Theorem 2. The latter result is proved in Sect. 3. Remark 2.5. The approach in this paper establishes weak mixing for geometric Lorenz semiflows. It then follows from Ratner [22] that such semiflows, and the corresponding flows, are Bernoulli. In particular, they are mixing. Suspensions and natural extensions. As promised, we end this section by recalling the notions of suspension and natural extension. Let (X, µ) be a probability space and f : X → X a (noninvertible) measure preserving transformation. Let r : X → R+ be
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an L1 roof function. Define the suspension X r = {(x, u) : x ∈ X, u ∈ [0, r(x)]}/ ∼ where (x, r(x)) ∼ (f x, 0). Form the suspension semiflow ft : X r → X r given by ft (x, u) = (x, u + t) computed modulo identifications. An invariant probability measure is given by µr = µ × / r dm, where is Lebesgue measure. Next suppose that ft : → is a semiflow preserving a probability measure ν. consisting of curves {x(s)}s≥0 in satForm the natural extension (or inverse limit) → is given isfying ft (x(s)) = x(s − t) for all s ≥ t ≥ 0. An invertible flow t : by t {x(s)}s≥0 = {x(s + t)}s≥0 . The projection π : → , {x(s)}s≥0 → x(0) defines a semiconjugacy between t and ft , and there is a unique t -invariant measure ν on such that π is measure preserving. 3. Mixing for Geometric Lorenz Flows In this section, we prove Theorem 2. Namely, we show that if f : I → I is an expanding map of Lorenz-type (satisfying f(i)–f(iii)) and r : I → R+ is a logarithmic roof function (satisfying r(i), r(ii)), then the corresponding geometric Lorenz flow is mixing. By Remark 2.5, it is enough to prove that the corresponding geometric Lorenz semiflow is weak mixing. Let f : I → I be a Lorenz-like expanding map. As shown in [9] and [10], f admits an induced map F that is Gibbs-Markov. More precisely there is an open interval Y ⊂ I containing 0 and (modulo a set of Lebesgue measure zero) there is a partition P = {ω} of Y consisting of intervals, and a return function R : Y → N constant on partition elements such that the induced map F (x) = f R(x) (x) restricts to a diffeomorphism F |ω : ω → Y for each partition element ω, and such that the following conditions are satisfied: (a) There exists λ > 1 such that |DFω | ≥ λ for each ω. (b) For all ω, log gω is H¨older (uniformly in ω), where g is the Jacobian of the inverse of F |ω : ω → Y . (c) R is Lebesgue integrable. A standard Folklore Theorem in dynamics says that conditions (a)–(c) imply the existence of a unique ergodic invariant measure µ for f : I → I that is absolutely continuous with respect to Lebesgue and whose support includes Y . In addition, the construction of [9, 10] satisfies (d) 0 ∈ f k ω for 0 ≤ k < R(ω) for each ω. The following Livˇsic regularity result is due to Bruin et al. [6]. Lemma 3.1. Suppose that f : I → I admits an induced map F : Y → Y satisfying conditions (a)–(d) above, with absolutely continuous ergodic measure µ. Let r : I → R+ be a roof function satisfying condition r(ii). Let ψ : I → S 1 be a µ-measurable function satisfying eir = (ψ ◦ f )ψ −1 , a.e. Then ψ has a version that is H¨older on Y . Sketch proof. We provide the details required for the reader to pass between the formulation in [6] and the statement of the lemma. By conditions (a)–(c), the map f : I → I can be modelled by a Young tower [32] with base Y . The appropriate measure for the
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tower has a H¨older density, so by condition (d), Bruin et al. [6, Theorem 2] applies to guarantee that ψ is H¨older on any fixed partition element of Y and hence on the whole of Y ([6, Remark 3]). The tower metric in [6] coincides with the Euclidean metric when restricted to Y , so ψ is H¨older in the original metric. Remark 3.2. Related Livˇsic regularity results can be found in [13]. The arguments in [6, 13] are different from each other leading to distinct results, and the approach in [6] turns out to more convenient for our purposes. Lemma 3.3. Suppose that the geometric Lorenz semiflow ft is not weak mixing. Then 1 there existska constant a > 0 and a measurable eigenfunction ψ : X → S continuous on k≥0 f Y , such that eiar = (ψ ◦ f )ψ −1 a.e.
(3.1)
Proof. It can be taken as the definition of weak mixing that φ ◦ ft = eiat φ has no measurable solutions φ : Xr → S 1 for a > 0. Suppose that φ is such a measurable solution. It follows from Fubini that there exists > 0 with r > a.e. such that φ ◦ ft (x, ) = eiat φ(x, ) for almost every x ∈ X. Set t = r(x) and ψ(x) = φ(x, ). Since fr(x) (x, ) = (f x, ) we obtain that ψ ◦ f = eiar ψ. Thus ψ is a measurable solution to Eq. (3.1). By Lemma 3.1, there is a solution ψ that is H¨older continuous on Y. It is now straightforward to show that ψ is continuous on k≥0 f k Y . Suppose that z = f k y, where y ∈ Y . Since f (0) is undefined, it is certainly the case that f j y = 0 for j = 0, . . . , k − 1. Hence we can choose an open set U ⊂ Y containing y such that 0 ∈ f j U for 0 ≤ j ≤ k − 1. In particular, there exists γ ∈ (0, 1) such that ψ is C γ on Y and at the same time eiark is C 1 on U . Let zi = f k yi where yi ∈ U for i = 1, 2. Applying Eq. (3.1) in the form ψ ◦ f k = iar e k ψ, we obtain ψ(z1 )ψ(z2 )−1 = ψ(y1 )ψ(y2 )−1 eiark (y1 ) e−iark (y2 ) , so by f(ii), |ψ(z1 )ψ(z2 )−1 | ≤ D|y1 − y2 |γ ≤ D(cλk )−γ |z1 − z2 |γ . Hence ψ is H¨older on f k U and so is certainly continuous at z as required.
Proof of Theorem 2. Suppose that ft is not weak mixing. By Lemma 3.3, there exists a > 0 and a measurable eigenfunction ψ : I → S 1 satisfying (3.1), such that ψ is k continuous on k≥0 f Y . Since f is l.e.o., ψ is continuous on (−1, 1). In particular, ψ is continuous at f (±1). Iterate equation (3.1) to obtain eiar eiar◦f = (ψ ◦ f 2 )ψ −1 .
(3.2)
We evaluate this equation along a sequence xn > 0 with xn → 0. We have the limits ψ(xn ) → ψ(0), ψ(f 2 xn ) → ψ(f (−1)), and r(f xn ) → r(−1). Hence, the right-hand-side of Eq. (3.2) and the second factor on the left-hand-side converge as n → ∞. To obtain a contradiction, it suffices to choose xn → 0 so that eiar(xn ) does not converge. For n sufficiently large, choose bn > r(/2n ) such that eiabn = (−1)n . Since r is continuous on I \ {0} and r(x) → ∞ as x → 0, there exists xn > 0 such that r(xn ) = bn . As required, xn → 0 and eiar(xn ) does not converge.
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4. Extensions and Future Directions In this paper, we have focused on geometric Lorenz attractors satisfying a locally eventually onto condition. In this section, we discuss how to relax the l.e.o. condition, and we describe how our results extend to larger classes of singular attractors. We end by mentioning some future directions. The l.e.o. assumption occurs in much of the literature since it is a verifiable property of the flow that guarantees topological transitivity of f . Moreover, it holds for the Lorenz equations (1.1) themselves [28]. However, topological transitivity for f : I → I is not crucial in this paper. We could simply work with the unique ergodic component whose support includes the set Y introduced in Sect. 3. Passing to the natural extension of the suspension over this component yields a geometric Lorenz attractor with an SRB measure as before. The proof of mixing for this attractor also relies to some extent on the l.e.o. property, but it is clear that the proof goes through under a much weaker condition: namely that a preimage of a forward image of 1 or −1 lies in Y . (That is, there exist integers k, ≥ 0 such that f (1) ∈ f k Y or f (−1) ∈ f k Y .) This condition is open and dense within the class of Lorenz-like expanding maps, and in principle, it can be verified by a finite computation. Next, we note that the specific structure of Lorenz-like expanding maps is not so crucial for the proof of mixing. Properties f(i)–f(iii) could be relaxed, or altered completely, provided f admits an induced map F : Y → Y satisfying conditions (a)–(d). The existence of a Gibbs-Markov induced map is a very general condition, see [7, 3]. It clearly holds for uniformly expanding maps and, we have made use of the recent work [9, 10] for Lorenz-like expanding maps. Condition f(ii) in the definition of the Lorenz-like expanding map corresponds to the eigenvalue condition λu > |λs | which is valid for the Lorenz attractor. The resulting class of geometric Lorenz attractors are sometimes called expanding. There are also contracting geometric Lorenz attractors [24] that arise when λu < |λs |. It seems likely that such attractors can be shown to be mixing using the ideas in this paper. In situations where Lemma 2.1 fails, it may not be possible to reduce to a onedimensional expanding map. However, it is plausible that the Poincar´e map P could be modelled by an “unquotiented” Young tower as in [31] to which the ideas in this paper might still be applicable. In this paper, we have established mixing. An interesting open question is to prove results on the speed of mixing. In a different direction, it would be interesting to derive statistical limit laws such as the central limit theorem and invariance principles for the Lorenz attractor. Acknowledgements. This research was supported in part by EPSRC Grant GR/S11862/01. IM is greatly indebted to the University of Houston for the use of e-mail, given that pine is currently not supported on the University of Surrey network.
References 1. Afra˘ımoviˇc, V.S., Bykov, V.V., Sil nikov, L.P.: The origin and structure of the Lorenz attractor. Dokl. Akad. Nauk SSSR 234, 336–339 (1977) 2. Allan, D.W.: On the behaviour of systems of coupled dynamos. Proc. Cambridge Philos. Soc. 58, 671–693 (1962) 3. Alves, J.F., Luzzatto, S., Pinheiro, V.: Lyapunov exponents and rates of mixing for one-dimensional maps. Ergodic Th. & Dyn. Syst. 24, 637–657 (2004)
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4. Anosov, D.V.: Geodesic flows on closed Riemann manifolds with negative curvature. Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder Providence, R.I.: American Mathematical Society, 1969 5. Bowen, R.: Mixing Anosov flows. Topology 15, 77–79 (1976) 6. Bruin, H., Holland, M., Nicol, M.: Livsic regularity for Markov systems. Ergodic Theory Dynam. Systems (2005) to appear 7. Bruin, H., Luzzatto, S., van Strien S.: Decay of correlations in one-dimensional dynamics. Ann. Sci. ´ Norm. Sup. 36, 621–646 (2003) Ec. 8. Bullard, E.C.: The stability of a homopolar dynamo. Cambridge Phil. Soc 51, 744–760 (1955) 9. Diaz-Ordaz, K.: Decay of correlations for non-H¨older observables for expanding Lorenz-like one-dimensional maps. (2005) is: To appear in Disc. And Cont. Dyn. Syst. 10. Diaz-Ordaz, K., Holland, M., Luzzatto, S.: Statistical properties of one-dimensional maps with critical points and singularities. (2005) Preprint 11. Field, M., Melbourne, I., T¨or¨ok, A.: Stability of mixing and rapid mixing for hyperbolic flows. http: www.maths.surrey.ac.uk/personal/st/I.Melbourne/papers/rapid.pdf 12. Glendinning, P., Sparrow, C.: Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps. Phys. D. 62, 22–50 (1993)Homoclinic chaos (Brussels, 1991). 13. Gou¨ezel, S.: Regularity of coboundaries for non uniformly expanding Markov maps. http://org/list/math.DS/0502458, 2005 is: To appear in Proc AMS. ´ 14. Guckenheimer, J., Williams, R.F.: Structural stability of Lorenz attractors. Inst. Hautes Etudes Sci. Publ. Math. 50, 59–72 (1979) 15. Hide, R.: Geomagnetism, ‘vacillation’, atmospheric predictability and deterministic chaos. In: ‘Paths of Discovery’, scripta varia, Vartican city: Pontifical Acad. of Sci. Press, 2005 16. Katok, A.(in collaboration with K. Burns): Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dynamical systems. Ergodic Theory Dynam. Systems 14, 757–785 (1994) 17. Lorenz, E.D.: Deterministic nonperiodic flow. J. Atmosph. Sci. 20, 130–141 (1963) 18. Morales, C.A., Pacifico, M.J., Pujals, E.R.: Singular hyperbolic systems. Proc. Amer. Math. Soc. 127, 3393–3401 (1999) 19. Morales, C.A., Pacifico, M.J.: Mixing attractors for 3-flows. Nonlinearity 14, 359–378 (2001) 20. Pesin, Ya.: Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Ergod. Th. & Dynam. Sys. 12, 123–151 (1992) 21. Plante, Joseph F.: Anosov flows. Amer. J. Math. 94, 729–754 (1972) 22. Ratner, M.: Bernoulli flow over maps of the interval. Israel J. Math. 31, 298–314 (1978) 23. Rikitake, T.: Oscillations of a system of disk dynamos. Proc. Cambridge Philos. Soc. 54, 89–105 (1958) 24. Rovella, A.: The dynamics of perturbations of the contracting Lorenz attractor. Bol. Soc. Brasil. Mat. (N.S.) 24, 233–259 (1993) 25. Sataev, E.A.: Invariant measures for hyperbolic mappings with singularities. Usp. Mat. Nauk. 47, 147–202, 240 (1992) (Russian) 26. Sparrow, C.: The Lorenz equations: bifurcations, chaos and strange attractors. Applied Mathematical Sciences, Vol. 41, Berlin: Springer Verlag, 1982 27. Tucker, W.: The Lorenz attractor exists. C. R. Acad. Sci. Paris S´er. I Math. 328, 1197–1202 (1999) 28. Tucker, W.: A rigorous ODE solver and Smale’s 14th problem. Found. Comput. Math. 2, 53–117 (2002) 29. Viana, M.: Dynamical systems: moving into the next century. In: Mathematics unlimited–2001 and beyond, Berlin: Springer, 2001, pp. 1167–1178 ´ 30. Williams, R.F.: The structure of Lorenz attractors. Inst. Hautes Etudes Sci. Publ. Math. 50, 321–342 (1979) 31. Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147, 585–650 (1998) 32. Young, L.-S., Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999) Communicated by G. Gallavotti
Commun. Math. Phys. 260, 403–443 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1413-7
Communications in
Mathematical Physics
Stability for Quasi-Periodically Perturbed Hill’s Equations Guido Gentile1 , Daniel A. Cortez2 , Jo˜ao C. A. Barata2 1 2
Dipartimento di Matematica, Universit`a di Roma Tre, Roma, 00146, Italy. E-mail: [email protected] Instituto de F´ısica, Universidade de S˜ao Paulo, Caixa Postal 66 318, S˜ao Paulo, 05315 970 SP, Brasil. E-mail: {dacortez, jbarata}@fma.if.usp.br
Received: 18 October 2004 / Accepted: 8 April 2005 Published online: 20 September 2005 – © Springer-Verlag 2005
Abstract: We consider a perturbed Hill’s equation of the form φ¨ +(p0 (t) + εp1 (t)) φ = 0, where p0 is real analytic and periodic, p1 is real analytic and quasi-periodic and ε ∈ R is “small”. Assuming Diophantine conditions on the frequencies of the decoupled system, i.e. the frequencies of the external potentials p0 and p1 and the proper frequency of the unperturbed (ε = 0) Hill’s equation, but without making any assumptions on the perturbing potential p1 other than analyticity, we prove that quasi-periodic solutions of the unperturbed equation can be continued into quasi-periodic solutions if ε lies in a Cantor set of relatively large measure in [−ε0 , ε0 ] ⊂ R, where ε0 is small enough. Our method is based on a resummation procedure of a formal Lindstedt series obtained as a solution of a generalized Riccati equation associated to Hill’s problem.
1. Introduction In the present work we will consider the one-dimensional Hill’s equation (for a standard reference, see [24]) with a quasi-periodic perturbation φ¨ + (p0 (t) + εp1 (t)) φ = 0 ,
(1.1)
where p0 and p1 are two real analytic functions, the first periodic with frequency ω0 and the latter quasi-periodic with frequency vector ω1 ∈ RD , for an integer D ≥ 1 (for notational details see Sect. 1.1). No further assumption is made on the equation, besides requiring that the real parameter ε is small and that the unperturbed equation (i.e. for ε ≡ 0) has a fundamental set of real quasi-periodic solutions. For p0 constant such an equation has been extensively studied, also in connection with the spectrum of the corresponding Schr¨odinger equation φ¨ + εV (ω1 t)φ = Eφ, with V analytic and periodic in its arguments; see for instance [11, 27, 12, 19, 29, 25]. We also mention the recent [5] and also [6], where some properties of the gaps and of
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the instability tongues have been investigated. Different perturbations of Hill’s equation, with a L1 perturbing potential, have been considered for instance in [26, 31, 32, 18]. We are interested in the problem of studying conservation of quasi-periodic motions for ε different from zero but small enough. Of course, Eq. (1.1) can be considered as arising from an autonomous Hamiltonian system with d = D + 2 degrees of freedom, described by the Hamiltonian H = 0 A + ω0 A0 + ω1 · A1 + εp1 (α 1 ) f (A, A0 , α, α0 ),
(1.2)
where (A, A0 , A1 , α, α0 , α 1 ) ∈ R × R × RD × T × T × TD are action-angle variables, and f and 0 depend on the periodic potential p0 . For instance if p0 is a constant, say p0 = 1, then the variables (A0 , α0 ) disappear, 0 = 1 and f (A, α) = 2A cos2 α. In general the change of variables leading to (1.2) is slightly more complicated, but it can be easily worked out; we refer for instance to [9, 10]. In such a case the function f is still linear in the action variables. Hence systems like (1.2) are not typical in KAM theory, because the perturbation does not remove isochrony. What one usually does is to study the behavior of the solutions, in particular to understand if they are bounded (quasi-periodic) or unbounded (linearly or exponentially growing), when varying the parameters characterizing the external potential. In the case of the Schr¨odinger equation this can be done for a fixed potential, by varying the energy, which represents an extra free parameter, and information can be obtained about the spectrum. In [10] this is done for bounded solutions, so that conditions on E are obtained characterizing the spectrum of the corresponding Schr¨odinger operator. Here we are interested in a different case, which has not been discussed in literature. More precisely we consider the case in which the potential is fixed, and the parameters of p0 are such that the fundamental solutions of the corresponding Hill’s equation φ¨ + p0 (t) φ = 0 are quasi-periodic (this means that we are inside the stability regions). Hence for ε = 0 we have d = D + 2 fundamental frequencies ω1 , ω0 , 0 , where 0 is the proper frequency of the unperturbed Hill’s equation. Then we want to study if the solutions remain quasi-periodic when the perturbation is switched on. Even when this occurs, one expects that the proper frequency of the system is changed as an effect of the perturbation. Since the system is in fact a perturbation of an isochronous one, and we have no free parameter to adjust, either the proper frequency is changed to some perturbation order or it is never changed (if disposing of an extra additive parameter E, like in the case of the Schr¨odinger equation, one can use such a parameter instead of ε to change the proper frequency of the system, as it would be possible in our case by requiring for the average of p1 to be non-zero). But to follow all the possibilities requires some careful analysis, which one can avoid by assuming some non-degeneracy condition on the perturbation in order to control the change of the frequencies. By non-degeneracy condition we mean the following: we shall see that in the forthcoming construction of the quasi-periodic solution we must check at each iterative step whether the average of some function depending on p1 vanishes or not, and according to such a property the algorithm has to be changed (for instance at the first step such a property is equivalent to having that the average of p1 vanishes or not). Hence one could impose some nondegeneracy condition of p1 by requiring that at some step the corresponding average is non-zero. On the contrary we do not want to impose any condition on the perturbing potential p1 . Degeneracy problems of this kind are known to be not easy to handle. An example is given by Herman’s conjecture in the case in which one has a system of N harmonic
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oscillators where no assumption is made on the coupling terms of order higher than two: in such a case the conservation of a large measure of invariant tori has been proved only for N = 2 [17], cf. also [28]. We can mention also Cheng’s results on the conservation of lower (N − 1)-dimensional tori for systems with N degrees of freedom [7, 8]. In both problems, nonlinear oscillators and lower dimensional tori, additional difficulties arise when one looks for a solution without making any assumptions on the perturbations. The involved difficulties are very similar to those of our problem, because in all cases they are due to the change of the frequencies. Also in Herman’s case, of course, the problem rather simplifies if one assumes a non-degeneracy condition on the perturbation, which allows to remove isochrony to first perturbation order, but solving the problem in the general case is still an open problem. Even if we never really thought about it, we think that Herman’s case is more difficult with respect to ours because in that case all frequencies can change, whereas in our case most of the frequencies are fixed and only one of them can change. To come back to our problem, we fix the unperturbed torus and study for which values of ε (small enough) such a torus is conserved. In particular we are interested in the dependence on ε of the conserved torus: we shall find that the torus will be defined for ε in a Cantor set of large relative measure, and for such values of ε the system turns out to be reducible, that is conjugated to a constant flow [22]. We shall see also that one can give a meaning to the perturbation series, through a suitable resummation, in an analogous way to what was done in similar contexts in [14, 13, 15]. As physical applications of (1.1) one could think of Hill’s equation for the motion of the Moon which is perturbed by the presence of the other planets (in the approximation in which the latter move in their Keplerian orbits, and only their influence on the Moon is taken into account). For an introduction to Hill’s problem in astronomy we can refer to the classical textbook by Szebehely [30]. We can also mention a paper by Avron and Simon [1], in which the theory of quasi-periodic Hill’s equation is applied to the problem of the rings of Saturn, even if that application is more in the spirit of quasi-periodic Schr¨odinger operators (in the sense that they consider a free parameter which has the role of energy in order to study the complex groove structure in the rings). Likely the stability problem for a many-body system consisting of Saturn, a test dust particle and Saturn’s satellites (considered as perturbations) could be studied by applying the theory developed here. We assume that (1.1) for ε = 0 has two linearly independent solutions (fundamental solutions) which are quasi-periodic. By Floquet’s theorem [24], there are two such solutions of the form φ1 (t) = ei0 t w1 (t) and φ2 (t) = e−i0 t w2 (t), with w1 and w2 both periodic with period 2π/ω0 and 0 ∈ R. We do not study directly Eq. (1.1). Rather, we shall write φ in terms of a suitable function u, for which a first order differential equation can be derived. Indeed by setting t g0 (t ) dt , φ0 (t) = const. exp i
t
Q(t) = exp −2i
0
g0 (t ) dt
,
0
where φ0 is a quasi-periodic solution of (1.1) for ε = 0, with rotation vector (ω0 , 0 ), where the proper frequency 0 is the average of g0 , and defining t g(t ) dt , φ(t) = φ0 (t) exp i 0
g(t) = iεQ(t)u(t),
(1.3)
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one finds that u has to solve the equation (see Sect. 2.2 for details) u˙ = R + εQu2 ,
R = p1 Q−1 ,
(1.4)
which is an ordinary differential equation which could be of interest by its own. The advantage of this procedure is that, as we will see, we will be able to look for a solution of (1.4) with the same rotation vector ω = (ω1 , ω0 , 0 ) of the unperturbed system, something which cannot be done for the full unperturbed system, as the proper frequency 0 is expected to change (as usually happens when perturbing an isochronous system). In principle one could also develop a resummation method directly for the original equation (1.1), but we do not expect any simplification with that approach. That such a solution u(t) exists can be shown, and this is the core of the paper, provided one assumes, besides an obvious Diophantine condition on ω, that ε is small enough, say |ε| ≤ ε0 , and belongs to a suitable Cantor set E of large relative measure in [−ε0 , ε0 ]. By the latter we mean that one has limε→0+ meas(E ∩ [−ε, ε])/2ε = 1, with meas denoting Lebesgue measure. To recover the solution φ(t) we have to express it in terms of u. By using the relations given in (1.3) one realizes that, first, the solution could be unbounded (if the imaginary part of the average g of g did not vanish), and, second, even if this did not occur, an extra frequency ε = 0 + g would appear in addition to the d frequencies already characterizing the model, which would sound strange. But one can check that both problems are spurious, as g turns out to be real and dependence on time of the function φ(t), which, in principle, could be through the variables ω1 t, ω0 t, 0 t, ε t (by construction), is indeed only through the variables ω1 t, ω0 t, ε t, as formally noticed in the case treated in [2]. In other words, the dependence on 0 t disappears, and this means that the maximal torus, which in absence of perturbation has rotation vector (ω1 , ω0 , 0 ), can be continued for ε ∈ E, and the last component of the rotation vector is changed into an ε-dependent quantity ε (that the other components cannot change is obvious by the form of the equations of motion). Hence the solution of (1.4) provides directly a perturbation expansion for the correction of the proper frequency of the system: indeed ε − 0 = g , and g is expressed in terms of the solution u. Another advantage of our technique is that we will be able to write asymptotic expansions for the solutions, in terms of divergent power series, which can provide an accurate description of the dynamics within any prefixed accuracy. More precisely, we do not prove that the series do not converge, as we have only bounds on the coefficients of the formal power series expansion. However, divergence of the series is very likely, cf. also [15], Sect. 7, for an analogous discussion. We can now state our results in the following theorem. Theorem 1.1. Let p0 : R → R be real analytic and periodic with frequency ω0 and such that the fundamental solutions of the corresponding Hill’s equation φ¨ + p0 (t) φ = 0 are quasi-periodic with a proper frequency 0 ∈ R. Let p1 : R → R be real analytic and quasi-periodic with frequency vector ω1 ∈ RD for some D ≥ 1. Define ω := (ω1 , ω0 , 0 ) ∈ Rd with d = D + 2 and assume that m · ω1 + nω0 + 20 = 0, ∀(m, n) ∈ ZD+1 and, moreover |ω · ν| ≥
C0 , |ν|τ
∀ν ∈ Zd \ {0} ,
for two fixed positive constants C0 > 0 and τ > d − 1 (Diophantine conditions). Then, there exists ε0 > 0 small enough and a Cantor set E ⊂ [−ε0 , ε0 ] of large relative
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measure in [−ε0 , ε0 ] such that, for all ε ∈ E, (1.4) admits a quasi-periodic solution of the form U˜ ν (ε)eiω·νt , u(t) = U (ωt; ε) = ν∈Zd
where the sum above is absolutely and uniformly convergent for all t ∈ R and all ε ∈ E. Moreover, for all ε ∈ E, the system (1.1) is reducible and it has a quasi-periodic solution of the form ˜ m,n (ε)ei(m·ω1 +nω0 )t , φ(t) = (ε t, ω1 t, ω0 t; ε) = eiε t (m,n)∈ZD+1
where, by denoting with · the average of a quasi-periodic function (that is the constant term in its Fourier expansion), ε := 0 + g = 0 + iε Qu is real, and the sum above is absolutely and uniformly convergent for all t ∈ R and all ε ∈ E. Finally, if g = 0 then E = [−ε0 , ε0 ] and ε reduces to 0 . 2 In particular the proof of the result will imply that the equation is reducible for ε ∈ E. This means that for all ε ∈ E the solution of the matrix equation X˙ = P (t)X corresponding to (1.1) can be written as X(t) = Y (ω0 t, ω1 t) eCt X0 for some constant matrix C; one says in such a case that the flow is conjugated to the constant flow eCt [22]. It would be interesting to study what happens for ε outside the set E (cf. the results proved for the case of the Schr¨odinger equation with p0 = 0 and other related models [12, 21, 23]). Note that the fundamental solutions of the linear differential equation (1.1) depend on the resummation method that we will introduce later, and as the latter is not uniquely defined the solutions themselves can not be proved to be unique. Furthermore we can not exclude in principle that other quasi-periodic solutions exist, possibly with the same asymptotic expansion (in fact no uniqueness result as for analytic or Borel-summable functions can be relied upon). All these issues are not exclusive of our work, rather they are a limitation of the method itself and it appears in other problems where it has been applied (as in [13–15]). In our case this in not really a problem as far as we are interested in the solutions of (1.1) for ε ∈ E, because what we really need is finding any set of fundamental solutions in order to write down the general solution. However, also the set E of allowed values of the perturbation parameter depends on the resummation method, and it can happen that by giving a different prescription a different set is obtained: in principle a value of ε which does not belong to E, hence which has been excluded in our resummation, can become allowed by changing the resummation procedure. Hence one can ask if there are values of ε which are excluded by any possible resummation method, and, if any, what happens for such values. The rest of this paper is devoted to the proof of the above theorem. We organize this work as follows: in Sect. 2 we motivate and discuss the Ansatz used to solve (1.1) and in Sect. 3 we introduce the tree representation of the perturbative coefficients obtained, which is the basis for the forthcoming analysis. Section 4 is devoted to the solution of the “zero mode” problem, which is essential for constructing a consistent quasi-periodic solution for (1.4). Next, Sect. 5 brings the core idea of this paper: the renormalization of the formal solution. This process is implemented through a multiscale decomposition of propagators and a suitable resummation technique. As described in Theorem 1.1, the result is a convergent quasi-periodic solution for (1.4), well defined in a Cantor set E of relatively large measure in [−ε0 , ε0 ]. Section 6 is
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devoted to the proof of some technical lemmas which are related to estimates on the so- called “self-energy values”. These lemmas are crucial in the proof of Theorem 1.1, which is essentially performed in Sect. 7, where convergence of the renormalized expansion is shown. Next, in Sect. 8 we provide estimates on the measure of the set E where the renormalized solution exists. It is shown that E is of relatively large measure in a compact set [−ε0 , ε0 ]. Finally, Sect. 9 completes the proof of Theorem 1.1 by analyzing properties of the renormalized expansion. Sect. 10 closes the paper by discussing the rather trivial situation where we cannot fix the zero modes as in Sect. 4. This is the situation where the proper frequency of the unperturbed Hill’s equation is unchanged when the perturbation is switched on, i.e. ε = 0 . 1.1. Basic notations. In this paper N will denote the set of positive integers, Z the set of all integers and R the set of real numbers. Note that 0 ∈ / N. For any n ∈ N, Zn (or n R ) is the Cartesian product of Z (or R) n times. The set T denotes the one-dimensional torus, i.e. T = R/2π Z. Tn is the n-dimensional torus. For any n ∈ N, Zn∗ is defined as Zn \ {0}, i.e. Zn∗ is Zn with the exception of the zero. The same applies to Rn . Vectors in Zn (or Rn ) will be denoted either by boldface or underlined characters. Boldface characters will be used to denote a vector in a certain dimension d, i.e. ω ∈ Rd , ν ∈ Zd . Underlined characters will be used to denote a vector in a certain dimension D < d, i.e. ω1 ∈ RD , m ∈ ZD . The scalar product in Rn will be denoted as usual by a dot: v·w := v1 w1 +· · ·+vn wn , for v, w ∈ Rn . The 1 -norm of a vector v = (v1 , . . . , vn ) ∈ Rn is |v| := |v1 |+· · ·+|vn |, where in the r.h.s. |·| denotes the usual absolute value in R (or C). The complex conjugate of z ∈ C will be denoted by z∗ . For any discrete set A we denote by |A| the number of elements of A. Given a periodic or, more generally, a quasi-periodic function f (with components of its rotation vector which are rationally independent), we denote by f the average of f , T 1 f := lim dt f (t) = f0 , T →∞ 2T −T where f0 is the constant term of the Fourier expansion of f [20]. The symbol 2 will be used at the end of the statement of a theorem, lemma or proposition and will be used at the end of a proof.
2. Perturbative Analysis In this section we will begin our perturbative analysis. We start from a given complex quasi-periodic solution for the unperturbed version of (1.1), i.e. for ε = 0, and search for a perturbative solution for the full equation that formally tends to this unperturbed solution as ε → 0. For this, we apply an exponential Ansatz, whose geometrical motivation we briefly discuss below, leading to a generalized Riccati equation (Eq. (2.8), ahead). In the core of this paper we prove that this generalized Riccati equation admits a quasi-periodic solution under suitable conditions on the frequencies and on the coupling parameter ε and, as we prove below, this implies quasi-periodicity of the perturbed solution of (1.1).
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However, as we shall see, boundness on the solutions of (2.8) will automatically imply stability on the associate solutions of Hill’s equation. This will become more clear with Proposition 2.3.
2.1. Unperturbed equation. The following elementary result presents some basic properties of complex quasi-periodic solutions of the unperturbed Hill’s equation that partially motivates the approach of Sect. 2.2. Proposition 2.1. Let p0 : R → R be an analytic periodic function with period T0 = 2π/ω0 , such that the equation ¨ + p0 (t) φ(t) = 0 φ(t)
(2.1)
has two non-trivial, real, analytic, quasi-periodic and independent solutions φa and φb . Then, the complex quasi-periodic solution φ0 (t) = φa (t) + iφb (t) can be expressed in the form φ0 (t) = exp (i0 t + iψ0 (t)) ,
(2.2)
where 0 ∈ R and ψ0 : R → C is an analytic periodic function with frequency ω0 . 2 Proof. Since the Wronskian W (t) = φa (t)φ˙b (t) − φb (t)φ˙a (t) is a non-vanishing constant, W (t) = W0 = 0, ∀t ∈ R, one has |W0 | ≤ |φa (t)| |φ˙b (t)| + |φb (t)| |φ˙a (t)| ≤ D(|φa (t)| + |φb (t)|), where D := max{supt∈R |φ˙a (t)|, supt∈R |φ˙b (t)|} < ∞, because φ˙a and φ˙b are both, by hypothesis, quasi-periodic. Let φ0 := φa +iφb . By the equivalence of the 1 and 2 norms, there exists a constant C > 0 such that |φ0 (t)| =
|φa (t)|2 + |φb (t)|2 ≥ C(|φa (t)| + |φb (t)|) ≥
C|W0 | , D
∀ t. (2.3)
This tells us that the quasi-periodic complex function φ0 remains outside of a neighborhood of the origin for all times. Under these circumstances, a theorem of H. Bohr [4], implies that we can write φ0 (t) = exp(i0 t + iψ0 (t)), where 0 ∈ R and ψ0 (t) : R → C is almost periodic. Floquet’s theorem guarantees that ψ0 is periodic with the same frequency of p0 . We clearly see from (2.2) that 0 is the rotation number of φ0 . Since φ0∗ is also a solution of (2.1) (because (2.1) is real), the most general (complex) solution is A1 exp +i0 t + iψ0 (t) + A2 exp −i0 t − iψ0 (t)∗ , (2.4) with A1 , A2 ∈ C. Defining the periodic function g0 (t) := ψ˙ 0 (t) + 0 , we can write t φ0 (t) = exp i g0 (t ) dt eiψ0 (0) . 0
Since ψ˙ 0 = 0, we have 0 = g0 .
(2.5)
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2.2. Perturbed equation and the exponential Ansatz. As we mentioned, the representation (2.5) is possible because (2.3) tells us that the quasi-periodic complex function φ0 runs outside of a neighborhood of the origin for all times. It is tempting to presume that this sort of stability property is preserved when the perturbation is switched on and that the periodic function g0 is replaced by a quasi-periodic one in the form g0 + g, where g vanishes when ε → 0. This is the motivation for the steps that follow. Let us now consider the perturbed equation (1.1) with p1 : R → R analytic and quasi-periodic, with frequencies in the set {m · ω1 , m ∈ ZD } for some D ≥ 1. The motivations presented above (see also [2]) lead us to search for a solution of (1.1) with the following form: t t φ(t) = φ0 (t) exp i = exp i g(t ) dt [g0 (t ) + g(t )] dt eiψ0 (0) , (2.6) 0
0
with g vanishing identically for ε = 0. It is easily verifiable that g must satisfy the following generalized Riccati equation: g˙ + ig 2 + 2ig0 g − iεp1 = 0 .
(2.7)
Remark 2.2. Of course in this way we are considering a solution which reduces to the first function in (2.4) for ε = 0. In the following we could also consider solutions continuing for ε = 0 the second function in (2.4), and the analysis would be the same. The idea now is to search for a quasi-periodic solution g for the above equation. In this case, φ(t) = exp (iε t + iψε (t)), where t ε := 0 + g
and ψε (t) := ψ0 (t) + g(t ) − g dt . 0
Note that, if such a g exists, ψε would be also quasi-periodic. However, in order to assure that φ is quasi-periodic we have to show that ε is a real number, which is the case iff g ∈ R. This is established by the following proposition that shows that if g is quasi-periodic, then φ is automatically stable, i.e. the Lyapunov exponent Im(ε ) vanishes. Proposition 2.3. Let us assume that (2.7) has a quasi-periodic solution g. Then the average of g is real, that is g ∈ R. 2 Proof. Write g0 = x0 + iy0 and g = x + iy. Note that g0 = 0 ∈ R, hence y0 = 0. One has i g˙ 0 − g02 + p0 = 0, whose imaginary part gives x˙0 = 2x0 y0 . Moreover, one has g˙ + ig 2 + 2ig0 g − iεp1 = 0 (Eq. (2.7)), whose real part is x˙ − 2xy − 2xy0 − 2yx0 = 0. Combining the two equations we obtain x˙ − 2xy − 2x0 y − 2xy0 + (−2x0 y0 + x˙0 ) = 0, hence x˙ + x˙0 − 2(y + y0 )(x + x0 ) = 0. By defining z = x + x0 the above equation becomes z˙ = f (t) z, where the function f (t) = 2(y(t) + y0 (t)) is bounded (and quasi-periodic), hence, by explicit integration, t z(t) = exp 2 [y0 (t ) + y(t )] dt z(0), 0
where z(0) = x0 (0) + x(0) = 0 (if z(0) = 0 then z(t) ≡ 0 for all t, hence x(t) = −x0 (t) for all t, which requires x0 (t) = x(t) ≡ 0 for all t, and this is not possible as x0 = 0 = 0, so that x0 (t) cannot vanish identically). On the other hand z(t) has to be a bounded quasi-periodic function, and this requires y0 + y = 0, so that one has y = 0.
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Therefore, we can establish that φ(t) given in (2.6) is quasi-periodic provided we find a quasi-periodic g. Further remarks on properties of φ will be discussed in Sect. 9. A slightly simpler version of the generalized Riccati equation (2.7) above was studied in [14] by a tree expansion method (see, e.g., [16] and references therein). So, the idea now is to try to adapt the analysis of [14] to the context of the problem posed here. Of course new problems are expected because no non-degeneracy assumption is made. First of all, let us rewrite the Riccati equation (2.7) as in [14]. Since φ0 = 0 for all t ∈ R, we define u(t) by g(t) = iεQ(t)u(t), where t = (φ0 (t))−2 , g0 (t ) dt Q(t) := exp −2i 0
which, by (2.2), is also quasi-periodic. We also define t g0 (t ) dt . R(t) := p1 (t)Q(t)−1 = p1 (t)φ02 (t) = p1 (t) exp 2i 0
With the above definitions one trivially checks from (2.7) that u˙ = R + εQu2 ,
(2.8)
which is very similar to the equation studied in [14]. The main difference with respect to [14] is that now no assumption is made on the perturbation.
3. Tree Expansion Now we pass to the perturbative expansions and a graphic representation that will conduct our analysis. As a first attempt (and also just to introduce notations) we search for a solution of (2.8) as a power series in ε: u(t) =
∞
ε k u(k) (t) .
(3.1)
k=0
Note that, in principle, u does not vanish identically for ε = 0, but g does, since g ∼ εu. By inserting the above Ansatz into Eq. (2.8), we arrive at u˙ (0) = R, u˙ (k) = Q
u(k1 ) u(k2 ) , ∀k ≥ 1 .
(3.2)
k1 +k2 =k−1
Since we search for a quasi-periodic solution u of (2.8), it is natural to introduce the following Fourier decomposition: iν·ωt u(k) , (3.3) u(k) (t) = ν e ν∈Zd
for some d ≥ 1 to be conveniently fixed later. Our goal now is to find a graphical (k) representation in terms of trees for the Fourier coefficients uν , as in [14].
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We now proceed and write the Fourier decomposition of the functions p0 , p1 , φ0 , Q and R. Since p0 is periodic with period T0 = 2π/ω0 , while p1 is quasi-periodic with spectrum of frequencies contained in the set {m · ω1 , m ∈ ZD }, we simply have p0 (t) = Pn(0) einω0 t , p1 (t) = Pm(1) eim·ω1 t . n∈Z
m∈ZD
We write the Fourier decompositions of φ02 and φ0−2 as follows: Fn(2) ei(nω0 +20 )t , Fn(−2) ei(nω0 −20 )t . (φ0 (t))2 = (φ0 (t))−2 = n∈Z
n∈Z
Therefore, the Fourier decomposition of R is Pm(1) eim·ω1 t Fn(2) ei(nω0 +20 )t = Rν eiν·ωt , R(t) = n∈Z
m∈ZD
ν∈Zd
where ν := (m, n1 , n2 ) , (1)
d := D + 2 ,
ω := (ω1 , ω0 , 0 )
(3.4)
(2)
and Rν := Pm Fn1 δn2 ,2 . With this notation, the Fourier decomposition of Q is as follows: Fn(−2) ei(nω0 −20 )t = Qν eiν·ωt , Q(t) = n∈Z
ν∈Zd (−2)
where ν, d and ω are as (3.4) and Qν := δm,0 Fn1 δn2 ,−2 . Remark 3.1. We assume the following non-resonant condition on the frequency vector ω: m · ω1 + nω0 + 20 = 0∀(m, n) ∈ ZD+1 . We also impose a Diophantine condition on ω, namely: |ω · ν| ≥
C0 |ν|τ
∀ν ∈ Zd∗ ,
(3.5)
with Zd∗ := Zd \ {0}, for two fixed positive constants C0 and τ > d − 1. Remark 3.2. By the analyticity assumption on p0 and p1 one obtains the following decay for the Fourier coefficients of Q and R: |Rν | ≤ Qe−κ|ν| ,
|Qν | ≤ Qe−κ|ν| ,
(3.6)
for some positive constants Q and κ. This will be essential in our forthcoming analysis. We now proceed and insert the decomposition (3.3) into (3.2). The result is the fol(k) lowing recursive relations for the coefficients uν , ν = 0: (iω · ν)u(0) ν = Rν , = (iω · ν)u(k) ν
k1 +k2 =k−1 ν 0 +ν 1 +ν 2 =ν
1 ) (k2 ) Qν 0 u(k ν 1 uν 2 , ∀k ≥ 1,
(3.7)
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for all ν = 0. Since the l.h.s. of (3.2) has zero average, one must also impose 0 = R0 , 0=
k1 +k2 =k−1 ν 0 +ν 1 +ν 2 =0 (1)
2 (k−1) 1 ) (k2 ) , Qν 0 u(k ν 1 uν 2 =: [Qu ]
∀k ≥ 1.
(3.8)
(2)
We note that R0 = P0 F0 δ0,2 = 0 so there is no problem with the requirement 0 = R0 . (k) The graphical representation of the coefficients uν is very similar to [14]. We give below the complete definitions with the end of making the exposition self-contained. Definition 3.3. A tree θ is a connected set of points and lines with no cycle such that all the lines are oriented toward a unique point called the root. We call nodes all the points in a tree except the root. The root only admits one entering line (called the root line). The orientation of the lines in a tree induces a partial ordering relation between the nodes: given two nodes v and w, we shall write w v if v is a long the path (of lines) which connects w to the root. We can identify in θ the following subsets: • E(θ ): the set of endpoints in θ. A node v ∈ θ will be an endpoint if no line enters v. • EW (θ ) ⊆ E(θ ): the set of white bullets in θ. With each v ∈ EW (θ ) we associate a mode label ν v = 0, an order label kv ∈ Z+ and a node factor Fv = α (kv ) . • EB (θ ) = E(θ) \ EW (θ ): the set of black bullets in θ . With each v ∈ EB (θ ) we associate a mode label ν v = 0 and a node factor Fv = Rν v . • V (θ): the set of vertices in θ. If v ∈ V (θ ), then v has at least one entering line. We associate with each vertex v ∈ V (θ ) a mode label ν v ∈ Zd and a node factor Fv = Qν v . • B(θ ) = EB (θ ) ∪ V (θ ): the set of black bullets and vertices in θ. • L(θ ): the set of lines in θ. Each line ∈ L(θ ) leaves a point v and enters another one which we shall denote by v . Since is uniquely identified with the point v which leaves, we may write = v . For each line we associate a momentum label ν ∈ Zd and a propagator g = 1/(iω · ν ) if ν = 0 and g = 1 if ν = 0; we say that the momentum ν flows through the line . The modes and the momenta are related by the following: if = v and , are the lines entering v, then νw . (3.9) ν = ν v + ν + ν = w∈B(θ) wv
We call equivalent two trees which can be transformed into each other by continuously deforming the lines in such a way that they do not cross each other. Definition 3.4. Let Tk,ν be the set of inequivalent trees θ satisfying: 1. for each vertex v ∈ V (θ ), there exist exactly two entering lines in v; 2. for each line which is not the root line one has ν = 0 if and only if leaves a white bullet; 3. one has |V (θ)| + v∈EW (θ) kv = k; 4. the momentum flowing through the root line is ν. We refer to Tk,ν as the set of trees of order k and total momentum ν.
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Based on the above definitions, we write for all k ≥ 0 and for all ν ∈ Zd , ν = 0: u(k) = ν
Val(θ ),
Val(θ ) :=
θ∈Tk,ν
g
∈L(θ)
Fv , (3.10)
v∈E(θ)∪V (θ)
where Val : Tk,ν → C is called the value of the tree θ and
g :=
1 iω·ν ,
ν = 0,
1 ,
ν = 0,
Fv
Qν v , v ∈ V (θ), := Rν v , v ∈ EB (θ ), α (kv ) , v ∈ E (θ ). W
(3.11)
(k)
All the trees which appear in the expansion of the coefficient uν belong to Tk,ν . Recip(k) rocally, every tree in Tk,ν appears in the graphical expansion of uν . It is clear that the (k) constants u0 = α (k) should be recursively fixed from conditions (3.8). We leave this for the next section. 4. Analysis of the Zero Modes. Fixing α (k ) , k ≥ 0 We now analyze Eqs. (3.7) and (3.8) in order to fix α (k) , k ≥ 0. One should keep in mind (0) that these equations are of a recursive nature. Therefore, one first starts by fixing uν , (1) ν = 0, from (3.7), then one fixes α (0) from (3.8), then one goes back to (3.7) to fix uν , ν = 0, and so on. Our intention here is to obtain a general recursive expression for the zero modes coefficients α (k) . We shall prove that, apart from a spurious situation, the only possible choice of constants α (k) compatible with (3.8) is α (k) = 0, for all k ≥ 0. Remark 4.1. Let θ ∈ Tk,ν , k ≥ 0, ν ∈ Zd . Since k = |V (θ)| + v∈EW (θ) kv , one clearly has 0 ≤ |V (θ)| ≤ k. If, e.g., EW (θ ) contains only one white bullet with order label k, then |V (θ)| = 0; on the other hand if EW (θ ) contains only white bullets with order label all equal to zero or if it is an empty set, then |V (θ )| = k. Another simple observation is that, by topological reasons, the total number of endpoints of θ is exactly |V (θ )| + 1 (this can be easily proved by induction). So, |EW (θ )| + |EB (θ )| = |V (θ )| + 1 and one has 0 ≤ |EB (θ )| ≤ |V (θ )| + 1. (k)
Lemma 4.2. In uν , k ≥ 0, ν = (m, n1 , n2 ) ∈ Zd , n2 belongs to the following set of even integers: {−2k, −2(k − 1), . . . , −2, 0, 2}. 2 (0)
Proof. For k = 0, n2 = 2 since uν ∝ δn2 ,2 . Now let k ≥ 1 and θ ∈ Tk,ν be a (k) tree contributing to uν . With each vertex v ∈ V (θ ) one associates the factor δn(v) ,−2 2 in Val(θ ) and with each black bullet b ∈ EB (θ ) the factor δn(b) ,2 . Thus, due to the 2 (v) conservation of momentum (3.9), one must have the constraint n2 = v∈V (θ) n2 + (b) b∈EB (θ) n2 = 2|EB (θ )| − 2|V (θ )| in the root line. From Remark. 4.1, one concludes that n2 = −2k, −2(k − 1), . . . , −2, 0, 2.
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Definition 4.3. Let θ ∈ Tk,ν , k ≥ 0, ν ∈ Zd , such that EW (θ ) is non-empty. Let Aθ ⊆ EW (θ ) be non-empty. We define θ \ Aθ as the Aθ -amputated tree generated by amputating the subset Aθ of white bullets from θ . This means that Val(θ \ Aθ ) = (kv ) −1 , where k Val(θ ) θ\Aθ = k − v∈Aθ α v∈Aθ kv denotes the order of the Aθ amputated tree. We call amputated line any line coming out from a white bullet in Aθ , (p) after amputation of Aθ . Now let Tk,ν := {θ ∈ Tk,ν : EW (θ ) = {v} with kv = p}. (p)
This means that a tree in Tk,ν has only one white bullet with order label p (and hence (p)
k − p vertices). We now amputate the white bullet in Tk,ν ; this gives the definition of (p) (p) (p) the set Tk,ν := {θ \EW (θ ) : θ ∈ Tk,ν }. Of course the order of a tree in Tk,ν is equal to its number of vertices, which is just k − p. We also introduce here the shorthand: (0) Tk,ν =: Tk,ν , for all k ≥ 1. Remark 4.4. From the previous definition and from the fact that g = 1 when leaves (p) a white bullet, one notes that Tk,ν = Tk−p,ν . (p) G Lemma 4.5. Let k ≥ 1, then [Qu2 ](k−1) = k−1 k−p , where, for all j ≥ 1, p=0 α 2 Gj := θ∈Tj,0 Val(θ ). Proof. By using the definition of [Qu2 ](k−1) , the definition of tree value in (3.10) and the notations of Definition 3.3 one can write
[Qu2 ](k−1) = Val(θ ) . θ∈Tk,0 (1)
(1)
(2)
(2)
Now let ν 1 = (m1 , n1 , n2 ) ∈ Zd and ν 2 = (m2 , n1 , n2 ) ∈ Zd . From Lemma 4.2, (1) (2) n2 ∈ {−2k1 , −2(k1 −1), . . . , −2, 0, 2} and n2 ∈ {−2k2 , −2(k2 −1), . . . , −2, 0, 2}. (k )
To be more precise, let θj ∈ Tkj ,ν j , j = 1, 2, be a tree contributing to uν jj , then (1)
(2)
n2 = 2(b1 − v1 ) and n2 = 2(b2 − v2 ), where bj and vj are the number of black bullets and the number of vertices in θj , respectively. From θ1 and θ2 we would like to construct a tree θ ∈ Tk,0 , k = k1 + k2 + 1, contributing to [Qu2 ](k−1) . First one must (0) note that the root lines of θ1 and θ2 enter a vertex in θ with mode ν 0 = (0, n1 , −2). As the line which exits this vertex (root line) carries zero momentum, one has the constraint (1) (2) −2 + n2 + n2 = 0. Thus, (b1 + b2 ) − (v1 + v2 + 1) = 0. This last relation implies that |EB (θ )| = |V (θ )|, so (see Remark 4.1) the tree θ contributing to [Qu2 ](k−1) must have exactly one white bullet (with some order label p). Of course |V (θ )| + p = k and 1 ≤ |V (θ)| ≤ k, hence 0 ≤ p ≤ k − 1. Therefore, one can write k−1 k−1
[Qu2 ](k−1) = Val(θ ) = α (p) Val(θ \ EW (θ )) p=0 θ∈T (p)
p=0 θ∈T (p)
k,0
=
k−1 p=0
α (p)
k,0
θ∈T k,0
(p)
Val(θ ) =
k−1 p=0
α (p)
k−p,0 θ∈T
Val(θ ),
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where Remark 4.4 was used. Hence the assertion is proved. Note that, by construction, Gj , j ≥ 1, is expressed by a sum of trees with no white bullets such that they have exactly j black bullets and j vertices. Definition 4.6. Let k ≥ 1 andTk,ν be as in Definition 4.3. We split Tk,ν into two disjoint c nc , where Tk,ν sets as follows: Tk,ν =: Tk,ν c : set of trees in T k,ν such that the amputated line is connected to the root line; • Tk,ν nc • Tk,ν : set of trees in Tk,ν such that the amputated line is not connected to the root line. c as a c-class tree and a tree in T nc as a nc-class tree. Note any We call a tree in Tk,ν k,ν nc-class tree has order k ≥ 2. nc can be transformed to be drawn in its “canonical form” as depicted in Any tree in Tk,ν nc , k ≥ 2, ν ∈ Zd , be arbitrary. Let v be the vertex connected Fig. 1. Indeed, let θ ∈ Tk,ν 1 to the amputated line of θ . Define v2 as the vertex such that one of its entering lines is exactly the line exiting v1 . Define vj inductively as the vertex such that one of its entering lines is exactly the line exiting vj −1 . If, for some j ≥ 2, vj is the vertex connected to the root line, then we set n := j . Now relabel the n vertices defined above as follows: vj = vn−j +1 , 1 ≤ j ≤ n. The vertices vj will be called canonical vertices. Set θj , 1 ≤ j ≤ n − 1, as the subtree whose root line is the one entering vj and not exiting vj +1 ; θn is defined as the subtree whose root line enters vn , not being the amputated line. The subtrees θj will be called canonical subtrees. Now draw the tree in such a way that the root line of each θj , 1 ≤ j ≤ n, is the upper line entering the vertex vj : in this way nc is as represented in Fig. 1. From now on, any tree in T nc is thought of as being θ ∈ Tk,ν k,ν drawn in its “canonical form”.
ν
v1
11 00 θ1 00 11 00 11 00 11 θ2 000 111 000 111 000 111 v2
... vn
11 00 θn 00 11 00 11 00 11
0
nc . The dashed bullet represents a general subtree containing Fig. 1. Canonical form of a tree in Tk,ν only black bullets. Each canonical subtree θj , 1 ≤ j ≤ n, is of order kθj and contains exactly kθj vertices and kθj + 1 black bullets. Taking into account the n canonical vertices {v1 , . . . , v2 }, one has k = n + nj=1 kθj . Note that 2 ≤ n ≤ k. The amputation of the white bullet leaves a line with vanishing momentum connected to the vertex vn ; we call such a line an amputated line
Stability for Quasi-Periodically Perturbed Hill’s Equations
417 (kθ )
Remark 4.7. Each canonical subtree θj ⊂ θ defined above gives a contribution to uν j j if kθj is the order of θj and if ν j is the momentum flowing through its root line. Of course ν j = 0 since this would give a contribution to α (kj ) and white bullets are discarded along the construction. Each line in L(θ ), which is neither the root line nor the line exiting from the amputated white bullet, can be seen as the root line of a subtree, hence it must have a momentum different from zero, as there are no other white bullets. nc admitting the same canonRemark 4.8. Note that there are 2n inequivalent trees in Tk,ν ical form with n canonical subtrees.
One now writes the value of a canonical subtree θj as (j )
Val(θj ) =
bν j
iω · ν j
(j )
≡ Bν j ,
ν j = 0 ,
1 ≤ j ≤ n.
t Therefore, θj gives a contribution to the function Bj (t) = 0 dt bj (t ) + Cj , where the integration constant Cj is chosen in such a way that summed to the constant term arising (j ) bν = from the definite integral gives the zero Fourier mode of Bj , that is Cj − ν∈Zd∗ iω·ν (j )
B0 . One can write Bj (t) =
(j )
Bν eiω·νt =:
bj ,
(4.1)
ν∈Zd∗
where the above integral has to be interpreted as a shorthand notation with the integration (j ) constant Cj fixed by imposing B0 = 0. One should think of it as just a zero average primitive of bj . Lemma 4.9. Let θ ∈ Tk,nc0 be a nc-class tree as the one in Fig. 1 with order k ≥ 2 and 2 ≤ n ≤ k canonical subtrees θ1 , . . . , θn . Let N0 := ν v ∈ Zd∗ : v ∈ B(θ ) and aj (t) := Q(t)Bj (t), 1 ≤ j ≤ n. Then (4.2) Val(θ ) = a1 a2 · · · an−1 an , N0
where all the integrals are in the sense of (4.1).
2
Proof. Let 1 ≤ j ≤ n and denote by ν 0,j the Fourier mode of the canonical vertex vj and by ν j the momentum flowing through the root line of the canonical subtree θj . Note that ν 0,j = 0, since this would give Q0 = 0. Also ν j = 0 and more generally, ν = 0, for all ∈ L(θ ) different from the root line and the line leaving the amputated white bullet (see Remark 4.7). Now the momentum flowing through the root line is zero, which means that nj=1 ν j + nj=1 ν 0,j = 0. Therefore, by an explicit computation, n (j ) n j =1 Qν 0,j Bν j Val(θ ) = n j j =1 r=1 ν r ν 0,r p=1 iω · ν n−p+1,0 + ν n−p+1 N0 = (QB1 ) (QB2 ) · · · (QBn−1 ) (QBn ) , (4.3) which proves the statement.
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Let θ ∈ Tk,nc0 be a nc-class tree as the one in Fig. 1 with order k ≥ 2 and n ≤ k canonical subtrees θ1 , . . . , θn . Of course if some two canonical trees θi , θj are equivalent, then one gets the same contribution in (4.2) by permuting ai with aj . This motivate us to give the following definitions: let = {θ1 , . . . θn } be the collection of all canonical subtrees of θ ∈ Tk,nc0 . We split into 1 ≤ m ≤ n disjoint subsets Ej , 1 ≤ j ≤ m, such that E1 is composed by all trees in which are equivalent to θ1 , E2 is composed by all trees in \ E1 which are equivalent to the first tree of \ E1 and so on. In this way, = m j =1 Ej , where each Ej collects together all trees which are equivalent to each other. Of course each subset Ej contains rj = |Ej | (equivalent) trees such that m j =1 rj = n. The contribution to (4.2) of all trees within the same Ej is denoted by aEj , where it represents the function ap associated to the tree θp which is equivalent to all trees in Ej . Now, let Sn denote the usual permutation group of n elements. We define Sn := Sn \ {π ∈ Sn : π(i) = j if i = j and θi , θj ∈ Ep for some 1 ≤ i, j, p ≤ n}. The set Sn will be called the set of all valid permutations within . Lemma 4.10. Let θ ∈ Tk,nc0 be a nc-class tree as the one in Fig. 1 with order k ≥ 2 and n ≤ k canonical subtrees θ1 , . . . , θn . Then (−1)n+1 d r1 (AE1 · · · ArEmm ) = 0 , aπ(1) aπ(2) aπ(3) · · · aπ(n) = r1 ! · · · rm ! dt π∈Sn
where AEj =
aEj , for all 1 ≤ j ≤ m.
2
Proof. First let us assume that all the subtrees θ1 , . . . θn are different. Therefore we have a1 = a2 = · · · = an in (4.2). With this assumption = {θ1 , . . . , θn } = nj=1 Ej with Ej = {θj } and rj = 1 for all 1 ≤ j ≤ n. By an integration by parts, one has a3 . . . an a2 a1 , a1 a2 a3 . . . an = − so that by summing also the term 1 ↔ 2 and performing another integration by parts, one obtains a1 a2 a3 . . . an + a2 a1 a3 . . . an d =− a3 . . . an (A1 A2 ) dt = a4 . . . an (A1 A2 a3 ) . Note that to construct the derivative of (A1 A2 ) above we have used the 2! = 2 permutations of a1 , a2 : (1 2 3 · · · n) and (2 1 3 · · · n). So, by using also (1 3 2 · · · n), (3 1 2 · · · n) and (3 2 1 · · · n), (2 3 1 · · · n), one gets and a4 . . . an (A3 A2 a1 ) . a4 . . . an (A1 A3 a2 ) Therefore, the sum of the 3! = 6 terms obtained by the permutation of a1 , a2 , a3 , gives d a5 . . . an (A1 A2 A3 a4 ) . a4 . . . an (A1 A2 A3 ) = − dt
Stability for Quasi-Periodically Perturbed Hill’s Equations
419
We now go on and sum the 4! = 24 terms obtained by the permutation of a1 , a2 , a3 , a4 to obtain a derivative of (A1 A2 A3 A4 ). We iterate this procedure until exhausting the n! permutations of a1 , . . . , an , giving d aπ(1) aπ(2) aπ(3) . . . aπ(n) = (−1)n+1 (A1 A2 . . . An ) = 0 , dt π∈Sn
which is the statement of the lemma in the case where all aj are different. Now assume the more general situation where = {θ1 , . . . , θn } = m j =1 Ej , for some m < n. Then we can permute i ↔ j iff ai = aj (we call this a valid permutation). The set of all valid permutations within is what we have denoted by Sn above. The n! total number of valid permutations is r1 !···r . Therefore, by using the result of the last m! formula, one arrives at the general statement. Remark 4.11. Note that the cancellation described by Lemma 4.10 occurs at fixed values of the mode labels. In other words, if we consider a fixed set of mode labels in N0 contributing to the sum in (4.3), and hence we replace each aj = QBj in the last line with the corresponding harmonic aj,ν j eiω·ν j t , we immediately realize that the argument given in the proof applies unchanged. Lemma 4.12. For all k ≥ 2 one has the identity j ≥ 1, Gj = θ∈T c Val(θ ).
nc θ∈T k,0
Val(θ ) = 0. Therefore, for all 2
j,0
Proof. Let k ≥ 2. The result follows by a combination of Lemma 4.9 and Lemma 4.10. Indeed, the sum of all possible trees in Tk,nc0 (including the sum over the Fourier modes) means that we have to sum all valid permutations of a1 , . . . , an in (4.2) for all trees with 2 ≤ n ≤ k canonical subtrees. Since this sum gives the average of a total derivative, one concludes that θ∈T nc Val(θ ) = 0. Now, since T1,0 contain only c-class trees, one k,0 concludes that Gj = Val(θ ) = Val(θ ) + Val(θ ) = Val(θ ) , j,0 θ∈T
c θ∈T j,0
nc θ∈T j,0
c θ∈T j,0
for all j ≥ 1. Proposition 4.13. Let Gj , j ≥ 1, be as the previous lemma. Suppose that Gj0 = 0 for some j0 ≥ 1. Then, (3.8) holds iff α (k) = 0 for all k ≥ 0. 2 Proof. By Lemma 4.5 condition (3.8) reads k−1
α (p) Gk−p , 0 = [Qu2 ](k−1) =
∀k ≥ 1.
(4.4)
p=0
We shall prove by induction that α (p) = 0, p ≥ 0, is the unique solution of (4.4) if Gj0 = 0 for some j0 ≥ 1. Indeed, let j0 ≥ 1 be such that G1 = · · · = Gj0 −1 = 0 and Gj0 = 0. Then, Eq. (4.4) is automatically satisfied for all 1 ≤ k ≤ j0 − 1. For j0 −1 (p) k = j0 , one has 0 = α (0) Gj0 + p=0 α Gj0 −p = α (0) Gj0 . Therefore, α (0) = 0. Now
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suppose that α (0) = · · · = α (k0 ) = 0 for some k0 ≥ 1 and let us prove that α (k0 +1) = 0. Using (4.4) for k = j0 + k0 + 1, we have 0=
j 0 +k0
α (p) Gj0 +k0 +1−p
p=0
=
k0
j 0 +k0
α (p) Gj0 +k0 +1−p +
α (p) Gj0 +k0 +1−p = α (k0 +1) Gj0 ,
p=k0 +1
p=0
which implies that α (k0 +1) = 0. Remark 4.14. Note that one can always suppose that the function p1 in (1.1) has zero (1) average (i.e. P0 = 0), by an appropriate choice of the average of p0 . In such a case, since one has (2) (1) Fn(−2) F−n1 1 G1 = Q R = 2 , (4.5) P0 i(20 − n1 ω0 ) n1 ∈Z
one finds G1 = 0. This shows that it is important to consider the possibility that the first non-vanishing Gj has j > 1. Proposition 4.15. Let Gj , j ≥ 1, be as the previous lemma. Then, k (a) Qu = 21 ∞ k=0 ε Gk+1 . (b) ε = 0 ⇔ Gk = 0 , ∀ k ≥ 1. (c) ε ∈ R ⇔ Gk = −Gk , ∀ k ≥ 1. In (a) the equality is in the sense of a formal power series (that is it holds order by order). 2 Proof. Let us first write Qu in Fourier space: Qu =
∞
ε k [Qu](k) ,
where
[Qu](k) =
ν 0 +ν 1 =0
k=0 (1)
(1)
Qν 0 u(k) ν1 ,
∀ k ≥ 1.
Now let ν 1 = (m1 , n1 , n2 ) ∈ Zd . Of course ν 1 = 0 since ν 0 = −ν 1 and Q0 = 0. (1) From Lemma 4.2, n2 ∈ {−2k, −2(k − 1), . . . , −2, 0, 2}. To be more precise, let (k) (1) θ1 ∈ Tk,ν 1 be a tree contributing to uν 1 ; then n2 = 2(b1 − v1 ), where b1 and v1 are the number of black bullets and the number of vertices in θ1 , respectively. From θ1 we would like to construct a tree θ ∈ Tk+1,0 contributing to [Qu](k) . We do this as follows. (0) Take the root line of θ1 entering a vertex v with mode ν 0 = (0, n1 , −2). Add a line, with zero momentum, entering such a vertex. We do not associate any propagator with this line, which means that it works as an amputated line (we call this the amputated line of θ); note that we can consider such a tree as a tree amputated of a white bullet. Finally, we let the root line of θ be the line exiting the vertex v carrying zero momentum. (1) Thus, ν 0 + ν 1 = 0. This last relation implies that −2 + n2 = 0, which means that b1 − (v1 + 1) = 0. Therefore, |EB (θ )| = |V (θ )|, leading to the conclusion that θ must
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421
have only one endpoint which is not a black bullet (see Remark 4.1). This leaves room only for the amputated line of θ, so thatθ1 must have only black bullets. Finally, one c concludes that if θ contributes to [Qu](k) , then θ ∈ Tk+1, 0 . On the other hand, only half c c (k) since in Tk+1, of the trees in Tk+1,0 contribute to [Qu] 0 we take into account two possibilities for amputating the leg connected to the root line. Therefore, by Lemma 4.12, one can write [Qu](k) =
1 2
c θ∈T k+1,0
Val(θ ) =
1 Gk+1 2
∞
and
Qu =
1 k ε Gk+1 , (4.6) 2 k=0
where the last formula holds as equality between formal series. This proves (a). Items (b) and (c) follow immediately from (a) remembering that ε = 0 + g = 0 + iε Qu .
5. Renormalization One of the main results of the last section (see Proposition 4.13) tell us that all α (k) = 0 if some Gj0 = 0, a condition which we henceforth assume; we shall come back to this in the last section. Therefore, one should not worry about white bullets and trivial propagators. For all k ≥ 0 and all ν ∈ Zd∗ , define Tk,ν := { θ ∈ Tk,ν : EW (θ ) = ∅ }. Then the expansion (3.10) still holds, with the definitions (3.11), provided one takes (k) EW (θ ) = ∅. Moreover, u0 = α (k) = 0 for all k ≥ 0.
Lemma 5.1. Let k ≥ 0 and ν ∈ Zd∗ , then |uν | ≤ AB k (k!)β e−κ |ν| , for positive constants A, B, β and κ < κ. 2 (k)
Proof. As in [14], p. 233, one can show that for all θ ∈ Tk,ν one has | Val(θ )| ≤ 1 2k (k!)β e−κ|ν|/4 v∈B(θ) e−κ|ν v |/4 , where 1 , 2 , β are suitable positive constants. Moreover the number of trees of fixed order is bounded by 3k , for some positive constant 3 . Thus, the assertion follows, with A = 1 , B = 2 3 and κ = κ/4. The main problem with the previous proof is that it does not treat conveniently the small denominators 1/ iω · ν which appear in the expansion through the propagators g . (k) As a result, we end up with a crude estimate for the coefficients uν , which complicates the task of studying the absolute convergence of the series for u. To overcome the problem of small denominators, we shall adopt a method well known from the analysis of the Lindstedt series for KAM type problems (see [16] and references therein). All the complication lies in the fact that ω · ν can be arbitrarily small for certain ν with sufficiently large |ν|. The idea, then, is to separate the “small” parts of ω · ν and to resume the corresponding terms in a suitable form, obtaining then a result which can be better estimated. The process of “separation” of the “small” parts of ω · ν is implemented via a technique known as the multiscale decomposition of the propagators. We stress that this technique is genuine from methods of the Renormalization Group introduced to deal with related problems in field theories.
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5.1. Multiscale decomposition of the propagators. As in [14] we introduce a bounded non-decreasing C ∞ (R) function ψ(x), defined in R+ , such that 1 , for x ≥ C1 , 0 , for x ≤ C1 /2 ,
ψ(x) =
where C1 ≤ C0 is to be fixed, with C0 the Diophantine constant which appears in (3.5), and setting χ (x) := 1 − ψ(x). An example of χ (x) and ψ(x) with the above properties is found [14], Fig. 5.1. We also define, for all n ∈ Z+ , χn (x) := χ (2n x) and ψn (x) := ψ(2n x). It is clear that χ0 (x) = χ (x), ψ0 (x) = ψ(x) and ψn (x)+χn (x) = 1, ∀ n ≥ 0. Functions χn (x) and ψn (x) allow us to write the propagator g , for all ∈ L(θ ) and ν = 0, as ∞
g =
(n) 1 = g , iω · ν n=0
with (0)
g :=
ψ0 (|ω · ν |) , iω · ν
(n)
g :=
ψn (|ω · ν |)χn−1 (|ω · ν |) , iω · ν
∀ n ≥ 1. (5.1)
(n)
We set g = g (n) (ω · ν ). Remark 5.2. Note that for fixed x = ω · ν, we have g (n) (x) = 0 only for two values of n. This means that the series (5.1) is, in fact, finite. Note also that g (n) (x) = 0 only if 2−n−1 C1 < |x| < 2−n+1 C1 for n ≥ 1 and only if |x| > 2−1 C1 for n = 0. Hence (n) (n) g = 0 ⇒ |g | ≤ C1−1 2n+1 . With each line ∈ L(θ ) with ν = 0 we associate a new label n = 0, 1, 2, . . . called the scale label of line . It is important to stress, based on Remark 5.2, that the scale (n ) label n of a line tells, essentially, what is the size of the associated propagator g . This is an useful device for “isolating” the contribution of trees containing propagators with too large scales. We shall do this carefully in what follows. Definition 5.3. We define k,ν as the set of trees which differ from those in Tk,ν by the introduction of the scale labels in the propagators. With the above definitions, expression (3.10) now reads as (n ) u(k) Val(θ ) , Val(θ ) = g Fv , ν = θ∈k,ν
∈L(θ)
(5.2)
v∈B(θ)
where the sum over all the trees in k,ν implies a further sum over all the possible scale labels for each one of the propagators. Thus, for all θ ∈ k,ν , if Nn (θ ) denotes the number of lines in θ on scale n, by using the bounds (3.6), Remark 5.2, and the fact that |L(θ)| = |B(θ )| = 2k + 1, we obtain ! ∞ " −1 n1 2k+1 −κ v∈B(θ ) |ν v | nNn (θ) , (5.3) e 2 | Val(θ )| ≤ (2C1 Q2 ) n=n1
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where we have introduced a (so far) arbitrary positive integer n1 and used the obvious fact that Nn (θ ) ≤ |L(θ )| = 2k + 1, ∀ n ≥ 0. Our problem now is to estimate Nn (θ ). To solve this, we need to introduce some useful definitions. Definition 5.4 (Cluster). A cluster T on scale n is a maximal connected subset of a tree θ such that all its lines have scale n ≤ n and there is at least one line on scale n. The lines entering a cluster T and the one (if any) exiting it are called the external lines of T . Given a cluster T on scale n, we denote by nT = n the scale of T . Moreover, V (T ), EB (T ), B(T ), and L(T ) denote, respectively, the set of vertices, black bullets, vertices plus black bullets, and lines contained in T ; the external lines of T do not belong to L(T ). We finally define the momentum of the cluster T as ν T := v∈B(T ) ν v . We shall call kT := |V (T )| the order of T . Some examples of clusters are presented in Fig. 2. Definition 5.5 (Self-Energy Graph). We call a self-energy graph any cluster T of a tree θ which satisfies out 1. T has only one entering line in T and only one exiting line T ; 2. The momentum of T is zero, i.e. ν T = v∈B(T ) ν v = 0. This means that ν in = ν out . T
T
We call a self-energy line any line out T
which exits from a self-energy graph T . We call a normal line any line which is not a self-energy line. Note that if T is a self-energy graph, out then in T , T ∈ L(T ), so that |L(T )| = 2kT − 1 and |B(T )| = 2kT . Some examples of self-energy graphs are depicted in Fig. 3. Remark 5.6. It is important to stress that due to the condition ν in = ν out , the scales on T T the entering and exiting lines of a self-energy graph T must differ at most by one unit,
T3
(2) (12)
(8)
(3) (4) (2)
(10) (8)
(4) (2)
(6)
(18)
ν T1
(11)
(1)
(10) (5)
(4)
(1) (3)
(11)
(3) (9)
(9)
(1)
(1) (3)
(1) (2)
(6)
(6)
(5)
T4 (3)
(1)
T5 T2
Fig. 2. Examples of clusters in a tree of order k = 16. The number between parentheses above each line denotes the scale of the propagator. Thus, we have nT1 = 3, nT2 = 1, nT3 = 8, nT4 = 2 and nT5 = 9. Note that T4 ⊂ T5 and therefore nT4 < nT5 . Of course there are other clusters in the example considered which are not shown
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G. Gentile, D. A. Cortez, J. C. A. Barata
T1
T2 T3
ν
ν,,
ν
T4
ν,,
ν ν, ν,
Fig. 3. Examples of self-energy graphs in a tree of order k = 7. Note that, in accordance with the definition, there is only one entering line and one exiting line (carrying the same momentum) in the self-energy graphs T1 , T2 , T3 and T4 . It is clear that the scales on the lines of T1 , T2 , T3 , T4 are strictly less than the scales on their external lines (after all, self-energy graphs are clusters)
i.e. |n in − n out | ≤ 1 (see Remark 5.2). Moreover, due to the fact that T defines a cluster, T T we must have nT + 1 ≤ min{n in , n out }, which is equivalent to saying that all the lines T
T
out within T have scale strictly less than the scale on the external lines in T and T .
Due to the presence of self-energy graphs one can have accumulation of small divisors. The heuristic explanation for this is as follows: imagine we have a line on a large scale n 1 entering a self-energy graph T . This line exiting from T could enter another self-energy graph T . Note that such a line is also on scale n 1. This process could repeat itself several times, resulting at the end in a bunch of lines 1 , . . . , N on scales n i 1, i.e. we end up with an accumulation of small divisors. Actually, from a more precise point of view, the whole problem with the self-energy graphs is that we are not able to give a satisfactory bound on the number of self-energy lines in a given tree θ . On the other hand if we denote by Nnnorm (θ ) the number of normal lines in a tree θ, then there exists a positive constant c such that |ν v | , (5.4) Nnnorm (θ ) ≤ c 2−n/τ v∈B(θ)
where τ is one of the Diophantine constants appearing in (3.5). Thus, suppose we could neglect all the self-energy lines within any tree θ , i.e. suppose that we could substitute Nn (θ) in (5.3) by Nnnorm (θ ) with the above estimate. Then, we would have | Val(θ )| ≤
(2C1−1 Q2n1 )2k+1 e−κ
v∈B(θ ) |ν v |
e
c log 2
∞
n=n1
n2−n/τ
v∈B(θ ) |ν v |
, (5.5)
for all n1 ≥ 0. Thus, picking n1 = n1 (κ, c, τ ) such that −
∞ κ + c log 2 n2−n/τ < 0 , 2 n=n
(5.6)
1
and summing over all the trees (whose number grows at most as k3 , for some positive 3 ) and all the Fourier labels, we would obtain | Val(θ )| ≤ 1 k2 k3 e−κ|ν|/4 , (5.7) |u(k) ν | ≤ θ∈k,ν
Stability for Quasi-Periodically Perturbed Hill’s Equations
425
what would imply in the convergence of expansion (3.1) provided |ε| < (2 3 )−1 . It is clear that the above result is false since we cannot simply forget the self-energy graphs. The estimate obtained just illustrates the fact that all the problem concerning the convergence of the series (3.1) lies in the existence of self-energy graphs (small divisors). We have to overcome this difficulty with some different approaches.
5.2. Renormalized expansion. The problem with the self-energy graphs can be solved by a suitable resummation procedure of the formal series obtained from the coefficients (n ) (5.2). The basic idea is to “dress” the propagators g in such a way that they could harbour all the malign contributions derived from the self-energy graphs. The next step is to define an expansion in terms of only non-self-energy graphs and renormalized propagators which we hope to give an estimate like (5.4). This is something analogous to the procedure of mass renormalization in field theories. We shall therefore iteratively define new propagators g [n ] (renormalized propagators). Definition 5.7 (Self-Energy Value). Suppose that the renormalized propagators g [n ] are given. For a self-energy graph T which does not contain any other self-energy graph, we define the self-energy value associated with T as VT (ω · ν; ε) := ε kT
g [n ]
∈L(T )
Fv ,
(5.8)
v∈B(T )
where ν is the momentum which enters T through the external line in T , kT = |B(T )|, and Fv is defined as in (3.11), with EW (θ ) = ∅. Note that VT (ω · ν; ε) depends on ω · ν through the propagators in L(T ). By setting x = ω · ν in (5.8) and x = ω · ν for each line ∈ L(T ), one can write x = x 0 + σ x, where x 0 = ω · ν 0 ,
ν 0 =
νw ,
(5.9)
w∈B(T ) wv : = v
and σ = 1 if is along the path of lines connecting the external lines of the self-energy T , and σ = 0 otherwise. Remark 5.8. The value VT (x; ε) of a self-energy graph T can depend on x only if kT ≥ 2. Definition 5.9. We define R k,ν as the set of renormalized trees, that is of trees which do R as the set of self-energy graphs of not contain any self-energy graph. We also define Sk,n order k which do not contain any other self-energy graph and such that the maximum of R is exactly n, and we call them the self-energy renormalized scales of the lines in T ∈ Sk,n graphs of order k and on scale n. We stress that the propagators associated with the lines [n ] R in R k,ν and Sk,n are the renormalized ones, g .
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Then we can define the renormalized propagators g [n] := g [n] (ω · ν ; ε) and the quantities M [n] (ω · ν ; ε) recursively as follows. For n0 ∈ Z+ we set M [n0 −1] (x; ε) := 0, M [n0 ] (x; ε) :=
g [n0 ] (x; ε) :=
∞
ψn0 (|x|) , ix
VT (x; ε),
k=1 T ∈S R
k,n0
M
[n0 ]
(x; ε) := χn0 (|x|) M [n0 ] (x; ε),
(5.10)
while, for n ≥ n0 + 1, by writing n (x; ε) := χn0 (|x|) . . . χn−1 (|ix − M[n−2] (x; ε)|) χn (|ix − M[n−1] (x; ε)|), n (x; ε) := χn0 (|x|) . . . χn−1 (|ix − M[n−2] (x; ε)|) ψn (|ix − M[n−1] (x; ε)|), we define g [n] (x; ε) :=
n (x; ε) , ix − M[n−1] (x; ε)
M [n] (x; ε) :=
∞
VT (x; ε),
k=1 T ∈S R
k,n
M (x; ε) := M (x; ε) + n (x; ε) M n = j (x; ε) M [j ] (x; ε), [n]
[n−1]
[n]
(x; ε) (5.11)
j =n0
where VT (x; ε) is defined as in (5.8). One should now realize, from the above definitions, that given an element in R k,ν R , all of its lines are on scale ≥ n . In particular, if T ∈ S R , all lines in T are or Sk,n 0 k,n0 exactly on the scale n0 and, hence, for all ∈ L(T ), the propagators are g [n0 ] (x ; ε), as in (5.10). Remark 5.10. Note that if a line is on scale n ≥ n0 + 1 and, by setting x = ω · ν , one has g [n] (x; ε) = 0, this requires χn0 (|x|) = 0, χn0 +1 (|ix − M[n0 ] (x; ε)|) = 0, . . . , χn−1 (|ix − M[n−2] (x; ε)|) = 0 and ψn (|ix − M[n−1] (x; ε)|) = 0, which means |ix − M[j ] (x; ε)| ≤ 2−(j +1) C1 , |ix − M
[n−1]
(x; ε)| ≥ 2
−(n+1)
n0 − 1 ≤ j ≤ n − 2,
C1 ,
so that, in particular, one has |g [n] | ≤ C1−1 2n+1 . If is on scale n0 and g [n0 ] (x; ε) = 0, then ψn0 (|x|) = 0, which implies that |g [n0 ] | ≤ C1−1 2n0 +1 . Then we define, formally, for ν = 0, u[k] ν =
θ∈R k,ν
Val(θ ),
Val(θ ) =
∈L(θ)
g [n ]
v∈B(θ)
Fv ,
(5.12)
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427
while, for ν = 0, one has u[k] 0 = 0, and we write u(t) =
∞
ε k u[k] (t) =
k=0
∞ k=1
εk
eiν·ωt u[k] ν ,
(5.13)
ν∈Zd
where the coefficients u[k] ν depend on ε (as the propagators do); note that the order k of a renormalized tree θ is still defined as k = |B(θ)|, but it does not correspond to the perturbative order any more. Definition 5.11. Let ω satisfy the Diophantine conditions (3.5). Fix ε such that one has # # # # #iω · ν − M[n] (ω · ν; ε)# ≥ C1 |ν|−τ1
∀ ν ∈ Zd∗ and ∀ n ≥ n0 ,
(5.14)
with Diophantine constants C1 and τ1 , where τ1 > τ and C1 < C0 are to be fixed later. We call E∗ the set of ε for which the Diophantine conditions (5.14) are satisfied, and we shall refer to it as the set of admissible values of ε. We shall see in the next section that for ε ∈ E∗ we shall be able to give a meaning to the (so far formal) renormalized expansion (5.13), hence we shall prove that the set E∗ has positive Lebesgue measure, provided that τ1 and C1−1 are chosen large enough. Fix ε such that the series obtained from (5.13) by replacing g [n ] in (5.12) with the bound 2n +1 C1−1 converges for |ε| ≤ ε, and fix ε0 ≤ ε small enough (how small will be determined by the forthcoming analysis). In the following we shall consider the interval [0, ε0 ]; the interval [−ε0 , 0] can be studied in the same way. It will be convenient to split the interval [0, ε0 ] into infinitely many disjoint intervals by setting [0, ε0 ] = {0} ∪
∞ $
% Em := 2−(m+1) ε0 , 2−m ε0 ,
Em ,
(5.15)
m=0
and to study separately each interval Em . We shall prove that for each m the admissible values of ε inside Em have large measure, and that their relative measure meas(Em ∩ E∗ )/meas(Em ) tends to 1 as m tends to infinity. Therefore in the following we imagine we have fixed m, and we set εm = 2−m ε0 , so that we can write Em = (εm /2, εm ].
6. Properties of the Self-Energy Values R define Given a self-energy T ∈ Sk,n 0
V T (x; ε) := εkT
∈L(T )
1 Fv , ix v∈B(T )
(6.1)
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G. Gentile, D. A. Cortez, J. C. A. Barata
which differs from VT (x; ε) as ψn0 (|x |) is replaced with 1 for all ∈ L(T ), and set [n0 ] 0] M[n j (x; ε) = χn0 (|x|)Mj (x; ε),
Mj[n0 ] (x; ε) =
j k=1
[n0 ]
[n ]
Mj (x; ε) = χn0 (|x|)M j 0 (x; ε),
[n ]
M j 0 (x; ε) =
j k=1
εk
εk
VT (x; ε),
R T ∈Sk,n 0
V T (x; ε). (6.2)
R T ∈Sk,n 0
[n0 ] [n0 ] [n0 ] 0] This allows us to decompose M[n j (0; ε) = Mj (0; ε)+ Mj (0; ε)−Mj (0; ε) , [n0 ]
0] where Mj (0; ε) depends neither on x nor on n0 . Note that one has M[n j (0; ε) =
[n0 ]
[n ]
Mj[n0 ] (0; ε) and Mj (0; ε) = M j 0 (0; ε) as χn0 (0) = 1 for all n0 ≥ 0. Lemma 6.1. Let Gj , j ≥ 1, be as the previous sections. Then one has [n0 ]
Mj0 (0; ε) =
j0
ε k Gk
(6.3)
k=1
2
for all n0 and all j0 . [n0 ]
R contributing to M Proof. By setting x = 0 any self-energy graph T in Sk,n j0 (x; ε) 0 looks like a tree θ in Tk,0 , except for the presence of the scale labels (compare (6.1) R each line ∈ L(T ) has scale n = n ). Nevertheless the with (5.8): if T ∈ Sk,n 0 0 corresponding propagators do not depend on the scales. Hence T ∈S R V T (0; ε) = k,n k,0 Val(θ ), so that the assertion follows from the definition of Gk (see Lemma 4.5). θ∈T
Lemma 6.2. For any self-energy T one has 1− 1 − χn0 (|x |) ≤ χn0 (|x |), ∈L(T )
∈L(T )
and the same result holds if each x is replaced with x 0 .
2
j −1 Proof. It follows from the identity nj=1 1 − aj = 1 − a1 − nj=2 aj i=1 (1 − ai ), with n ≥ 2 and 0 ≤ aj ≤ 1, which can be easily proved by induction. Lemma 6.3. For all n0 one has # # n0 /τ1 [n0 ] # [n0 ] # , #Mj0 (0; ε) − Mj0 (0; ε)# ≤ B1 |ε| e−B2 2 # # n0 /τ1 [n0 ] # # [n ] , #∂ε Mj0 0 (0; ε) − Mj0 (0; ε) # ≤ B1 e−B2 2 for suitable constants B1 and B2 , depending on j0 but independent of n0 .
2
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429
Proof. One can write M [n0 ] (0; ε) as in (5.10) with x = 0, where VT (0; ε) is given by (5.8) with n = n0 and g [n ] = ψn0 (|x 0 |)/ ix 0 = (1 − χn0 (|x 0 |))/ ix 0 . Furthermore [n0 ]
0] M[n j0 (x; ε) and Mj0 (0; ε) are polynomials of degree j0 in ε, hence one has
[n0 ] 0] M[n j0 (0; ε) − Mj0 (0; ε)
=
j0
VT (0; ε) − V T (0; ε) ,
k=1 T ∈S R
k,n0
which is trivially differentiable with respect to ε. By applying Lemma 6.2, we obtain # # [n0 ] # # [n ] # Mj0 0 (0; ε) − Mj0 (0; ε) # ≤
j0 k=1
|ε|k
R ∈L(T ) T ∈Sk,n
(6.4)
1 |Fv | , 0| |x ∈L(T ) v∈B(T )
χn0 (|x 0 |)
0
[n0 ]
k−1 0] and for ∂ε (M[n j0 (0; ε) − Mj0 (0; ε)) the same bound can be obtained with k |ε| 0 0 k −n instead of |ε| . In (6.4) the factor χn0 (|x |) requires |x | ≤ C1 2 0 , so that by the Diophantine condition (3.5) one has |ν 0 | ≥ 2n0 /τ , hence in v∈B(T ) |Fv | ≤ −κ|ν v |/2 −κ|ν v |/4 −κ|ν v |/4 one can bound the Q|B(T )| v∈B(T ) e v∈B(T ) e v∈B(T ) e 0 n /τ n /τ third product by v∈B(T ) e−κ|ν v |/4 ≤ e−κ|ν |/4 ≤ e−κ2 0 /4 < e−κ2 0 1 /4 , if τ1 > τ0 , while using the second product to perform the sum over the mode labels and the first one to find, by reasoning as for the proof of Lemma 5.1,
τ ! 2|L(T )| τ |L(T )| 1 k −κ|ν v |/2 ≤ e ≤ 1 2T (kT !)β , 0 C0 κ |x | ∈L(T ) v∈B(T )
R , so that, by collecting together the bounds and inserting them with kT = k for T ∈ Sk,n 0 j
into (6.4), we prove the assertion. In particular B1 is proportional to 20 (j0 !)β , while B2 is independent of j0 . Lemma 6.4. Let Gj , j ≥ 1, be as the previous sections. Assume that there is j0 ∈ N such that Gj0 = 0 and Gj = 0 for all 1 ≤ j < j0 . There exists two constants c1 and c2 , depending on j0 , such that for 1 n0 ≥ τ1 log2 c1 + c2 log |ε|
(6.5)
# # j 0 # # [n ] |ε|j0 −1 |Gj0 |, #∂ε Mj0 0 (0; ε)# ≥ 2
(6.6)
one has
provided ε is small enough. If j0 = 1 one can take c2 = 0.
2
430
G. Gentile, D. A. Cortez, J. C. A. Barata [n0 ]
Proof. One can write Mj0 (0; ε) = [n0 ] ∂ε Mj0 (0; ε)
j0
k=1 ε
kG
k
= ε j0 Gj0 , by Lemma 6.1, so that
= j0 ε j0 −1 Gj0 . By Lemma 6.3, we can bound
# # # # 1 n0 /τ1 [n0 ] # # [n ] ≤ |ε|j0 −1 #Gj0 # , #∂ε Mj0 0 (0; ε) − Mj0 (0; ε) # ≤ β1 B1 e−β2 B2 2 2
(6.7)
where the first inequality is obtained as soon as β2 ≤ 1 and β1 ≥ 1, while the second one requires 1 2β1 B1 1 j0 − 1 log log n0 ≥ τ1 log2 + , (6.8) β2 B 2 |Gj0 | β2 B 2 |ε| so that the assertion follows if c1 and c2 are chosen according to (6.8). Remark 6.5. The constants β1 and β2 in (6.7) could be taken β1 = β2 = 1. However in the following it will turn out to be useful to have some freedom in fixing their values; see in particular Remark 7.6. Remark 6.6. Note that if we choose n0 = τ1 log2 (c1 + c2 log(2/εm )) we obtain a value of n0 which can be used for all ε ∈ Em .
7. Convergence of the Renormalized Expansion We are left with the problem of proving that the series defining the renormalized expansion (5.13) converges, and of studying how large is the set E∗ ∩ [0, ε0 ] of admissible values of ε; we shall verify that it is a set with positive relatively large measure. As we have fixed m, for notational simplicity, in the following we shall find it convenient to shorthand E [∞] ≡ E∗ ∩ Em . We shall assume ε ∈ E [∞] , and n0 fixed as in Remark 6.6. Lemma 7.1. Assume that the set E [∞] has non-zero measure and that for all ε ∈ E [∞] and for all n0 ≤ j < n − 1 the functions M[j ] (x; ε) are C 1 in x and satisfy the bounds # # # # # # # # [j ] (7.1) #∂x M[j ] (x; ε)# ≤ D |ε|, #M (x; ε)# ≤ D |ε|, for some constant D. There there exists a positive constant c, independent of n, such that for any renormalized tree θ with Val(θ ) = 0 the number Nj (θ ) of lines on scale j satisfies the bound |ν v |, (7.2) Nj (θ ) ≤ c 2−j/τ1 v∈B(θ)
for all n0 < j ≤ n − 1.
2
Proof. The proof is the same as that of Lemma 1 in [14]. Just note that in [14] the bound |M[j ] (x; ε)| ≤ √ D|ε| is used only for j < n − 1 and the argument still applies if we replace |ε| with |ε| in the bound. At the end one obtains c = 2 23/τ1 .
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431
Remark 7.2. Let j0 be as in Lemma 6.4. If ε0 is small enough, for all ε ∈ (εm /2, εm ] and n0 chosen according to Remark 6.6, if j0 = 1 we can bound (2k−1)τ1 |ε|k , while if j0 > 1 we can bound |ε|k 2(2k−1)n0 ≤ |ε|k 2(2k−1)n0 ≤ c1 (2c2 )(2k−1)τ1 |ε|k (log(2/|ε|)(2k−1)τ1 , where, under the same smallness assumption on ε, one has (log(2/|ε|))τ1 ≤ Sp (1/|ε|)p ,
(7.3)
for all p > 0 and with Sp a positive constant depending on p. Hence, by taking p ≤ 1/4, one obtains, for j0 > 1, |ε|k 2n0 (2k−1) ≤ (2c2 )(2k−1)τ1 Sp2k−1 |ε|k/2 . Therefore, whichever the value of j0 is, we can bound |ε|k 2(2k−1)n0 ≤ c3 |ε|k/2 ,
(7.4)
for all k ≥ 1, with c3 a suitable positive constant. Remark 7.3. In particular one can choose p ≤ 1/(2(2j0 + 1)), which implies |ε|k−1 (log(2/|ε|))(2k−1)τ1 ≤ |ε|j0 −1 |ε| |ε|(k−j0 −1)/2 for all k ≥ j0 + 1, a property which will be useful in the following. Lemma 7.4. Fix p as in Remark 7.2. Then one has # # # # # [n0 ] # # # #M (x; ε)# ≤ D |ε|, #∂x M[n0 ] (x; ε)# ≤ D |ε|, for suitable positive constants D and D .
2
Proof. The first bound follows from (7.4). Let j0 be as in Lemma 6.4. If j0 = 1 then n0 does not depend on ε, and also the second bound is trivially satisfied. If j0 ≥ 2, in order to obtain the second bound, one can discuss in different ways contributions with kT = 1 and contributions with kT ≥ 2. If kT = 1 then VT (x; ε) does not depend on x (see Remark 5.8), so that, by using the notations (6.2), one has [n0 ] [n0 ] [n0 ] [n0 ] M 1 (x; ε) = χn0 (|x|)M1 (0; ε), and one can write M1 (0; ε) = M 1 (0; ε) + [n ]
[n ]
M1[n0 ] (0; ε) − M 1 0 (0; ε) , where M 1 0 (0; ε) = 0 by Lemma 6.1 (and the defini[n0 ]
[n0 ] 0] tion of j0 ), while the difference M[n 1 (0; ε) − M1 (0; ε) = χn0 (|x|) (M1 (0; ε) − [n ]
M 1 0 (0; ε)) can be bounded through Lemma 6.3 proportionally to e−B2 2 0 1 . Hence the derivative with respect to x acts only on the compact support function χn0 (|x|) and n /τ produces a factor 2n0 which is controlled by the exponentially small factor e−B2 2 0 1 . The conclusion is that the contributions with kT = 1 can be bounded proportionally to ε. The contributions with kT = 2 can be bounded relying again on the bound (7.4). n /τ
Lemma 7.5. Fix p as in Remark 7.2 and n0 as in Remark 6.6. For ε ∈ E [∞] and for x such that g [n] (x; ε) = 0, there exist two constants D and D such that the functions M[j ] (x; ε) are smooth functions of x and satisfy the bounds # # # # # [j ] # # # #M (x; ε)# ≤ D |ε|, #∂x M[j ] (x; ε)# ≤ D |ε|, # # j/τ1 # [j ] # (7.5) #M (x; ε) − M[j −1] (x; ε)# ≤ D |ε|e−D 2 ,
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for all n0 < j ≤ n − 1. Furthermore for all T contributing to M[j ] (x; ε), with n0 < j ≤ n − 1, one has |ν v |, (7.6) Nj (T ) ≤ c 2−j /τ1 v∈B(T )
for all j ≤ j .
2
Proof. The first bound in (7.5) can be proved by induction on n0 ≤ j ≤ n − 1. For j = n0 it has been already checked (see Lemma 7.4). Let us assume that it holds for all n0 ≤ j < j . One can proceed as for the proof of Lemma 2 in [14]. First of all one can R the inequalities prove for any self-energy graph T ∈ Sk,j
|ν v | > 2(j −4)/τ1 ,
v∈B(T )
Nj (T ) ≤ 2 2(3−j )/τ1
|ν v | ,
n0 + 1 ≤ j ≤ j,
(7.7)
v∈B(T )
where Nj (T ) denotes the number of lines on scales j contained in T . We omit the proof, as it is identical to that given in [14]. The estimates (7.7) allow us to bound j/τ1 |VT (x; ε)| ≤ |ε|k A1 Ak2 e−A3 2 e−κ|ν v |/2 . (7.8) v∈B(T )
The only difference with respect to the analogous bound (7.18) in [14] is that the constants A1 and A2 depend on ε. In fact given a self-energy graph # [n T] # ∈ R Sk,j , if we express its value according to (5.8), we can bound ∈L(T ) #g # ≤ |L(T )| j 2n0 Nn0 (T ) n=n0 +1 2nNn (T ) , with Nn0 (T ) ≤ 2k − 1 and Nn (T ) ≤ 2C1−1 c 2−n/τ1 v∈B(T ) |ν v | for all n0 + 1 ≤ n ≤ j , as it follows from the second bound in (7.7). Hence the last product can be bounded by using the bound on Nn (T ) and (5.6) with n1 (κ, c, τ ) = n0 : just note that for ε0 small enough such a choice for n1 (κ, c, τ ) automatically satisfies the inequality in (5.6). Then we can apply the bounds given in Remark 7.2 to write k
|ε|k A1 Ak2 = A1 A2 |ε|k/2 ,
(7.9)
with A1 and A2 two constants independent of ε. Then the first bound in (7.5) is proven. To obtain the third bound in (7.5) we note that one has for j ≥ n0 + 1, ∞ # # # # # [j ] # # [j ] # [j −1] (x; ε)# ≤ #M (x; ε)# ≤ |VT (x; ε)|, (7.10) #M (x; ε) − M k=1 T ∈S R
k,j
where sum of the contributions with k ≥ 2 can be bounded proportionally to |ε| e−A3 2 1 , because of (7.8) and (7.9), while the contributions with k = 1 can be bounded proportionally to |ε| if j0 = 1, whereas if j0 ≥ 2 we can reason as follows. We can bound |VT (x; ε)| j/τ j/τ j/τ j/τ according to (7.8), with k = 1, and write e−A3 2 1 = e−A3 2 1 /2 e−A3 2 1 /4 e−A3 2 1 /4 . j/τ
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R ), hence The self-energy graph T contains exactly one line on scale j (as T ∈ S1,j
n = j and |g [n ] | ≤ C1−1 2j +1 , so that we can use that 2j e−A3 2 1 /4 is bounded by a conj/τ n /τ stant. Moreover we have e−A3 2 1 /4 ≤ e−β2 B2 2 0 1 ≤ |Gj0 |(2β1 B1 )−1 |ε|j0 −1 if β2 in (6.7) is chosen such that β2 B2 ≤ A3 /4 (see Remark 6.5). Therefore, we can conclude that j/τ if j0 ≥ 2 the first sum in (7.10) can be bounded proportionally to |ε||ε|j0 −1 e−A3 2 1 /2 . Hence the third bound in (7.5) follows for any value of j0 , with D = A3 /2. The second bound in (7.5) again can be proved by reasoning as in [14] for the contributions arising from self-energy graphs T with kT ≥ 2. The contributions arising from n/τ self-energy graphs T with kT = 1 can be bounded as |ε|QC1−1 2n+1 e−A3 2 1 e−κ|ν v |/2 (as in the bound on the first sum in the r.h.s. of (7.10)) because there is only one propagator on scale n. Then, if the derivative acts on the compact support function χn0 (|x|), n/τ one has that 2n0 2n+1 e−A3 2 1 /2 is bounded by a constant for all n > n0 . j/τ
Remark 7.6. As suggested by the proof of Lemma 7.5 we shall fix β2 in (6.7) such that one has D ≥ 2β2 B2 , where D is the constant appearing in the last of (7.5). For future convenience we shall choose β2 such that D = 4β2 B2 ; see (8.17). We shall see below that it will be useful (even not necessary) also to choose β1 in (6.7) such that 2D ≤ β1 B1 . Proposition that the set E [∞] has non-zero measure. Then for all ε ∈ E [∞] # Assume # [n ]7.7. −1 n +1 # # ≤ C1 2 one has g for all lines in any tree or self-energy graph. In particular the series (5.13) is uniformly convergent to a function analytic in t. 2 Proof. It follows from Lemma 7.1, by taking the limit n → ∞ and using that the constant c does not depend on n, that the bound (7.2) holds for all j > n0 . Then one can bound the product of propagators as done in the proof of Lemma 7.5, and using part of the decaying factors e−κ|ν v | to obtain an overall factor e−κ|ν|/4 for any tree θ ∈ k,ν contributing to u[k] ν . 8. Measure of the Set of Admissible Values To apply the above results we have still to construct the set E∗ for which the Diophantine conditions (5.14) hold, and to show that such a set has positive measure. Here and henceforth we assume that the constants n0 and p are chosen according to Remark 6.6 and Remark 7.3, respectively. Define recursively the sets E [n] as follows. Set E [n0 ] = Em and, for n ≥ n0 + 1, & ' E [n] := ε ∈ E [n−1] : |iω · ν − M[n−1] (ω · ν; ε)| > C1 |ν|−τ1 , (8.1) for suitable Diophantine constants C1 and τ1 (to be fixed later). It is clear that E∗ ∩ Em = E [∞] =
∞ ( n=n0
E [n] = lim E [n] . n→∞
(8.2)
Lemma 8.1. The functions M [n] (x; ε) and their derivatives ∂x M [n] (x; ε) are C 1 extendible in the sense of Whitney outside E [n−1] , and for all ε, ε ∈ E [n−1] one has ∂xs M [n] (x; ε ) − ∂xs M [n] (x; ε) = ε − ε ∂ε ∂xs M [n] (x; ε) + o ε − ε , (8.3)
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where s = 0, 1 and ∂ε ∂xs M [n] (x; ε) denotes the formal derivative with respect to ε of ∂xs M [n] (x; ε). Furthermore one has # # # # n/τ1 # # # # (8.4) #∂ε ∂x M[n] (x; ε)# ≤ D |ε|, #∂ε ∂x M [n] (x; ε)# ≤ D |ε|e−D 2 , 2
for all n > n0 . One can take D as in Lemma 7.5.
Proof. Similar to the proof of Lemma 3 in [14]. In order to obtain the inequality (8.4) one has to use Remark 5.8. Of course, when expressing M[n] (x; ε) in terms of the selfenergy values VT (x; ε) we have to bear in mind that the constant 2n0 can be bounded in terms of ε, but it does not depend on ε (as far as ε varies in Em and n0 is chosen according to Remark 6.6), so that the derivatives with respect to ε of VT (x; ε), as expressed in (5.8), act only on εkT and on the quantities M[j ] (x; ε) appearing in the propagators. Hence ∂ε VT (x; ε) and ∂ε ∂x VT (x; ε) can be studied as in [14]. We simply note that when acting on some propagator g [n ] the derivatives with respect to ε can raise the power of the divisor ix − M[n −1] (x; ε), and if n = n0 we have to use part of the exponential j/τ decay e−A3 2 1 (see (7.8)) to take into account the extra factors 2n0 . The conclusion is that essentially the derivative with respect to ε of VT (x; ε) admits the same bound (7.8) as VT (x; ε), possibly with different constants A1 and A2 (but still such that a bound like (7.9) is fulfilled, as far as their dependence on ε is concerned), except that the exponent of |ε| is k − 1 instead of k. Therefore for all ε ∈ E [n−1] the quantities M[n] (x; ε) are well defined and formally differentiable (in the sense of Whitney) together with their derivatives with respect to x. Lemma 8.2. There are two positive constants ᒊ1 and ᒊ2 such that # # # # #∂ε M[n] (x; ε)# ≥ ᒊ1 |ε|j0 −1 − ᒊ2 |ε| |x| , for all n ≥ n0 .
(8.5) 2
Proof. If we write
x
∂ε M (x; ε) = ∂ε M (0; ε) + [n]
[n]
dx ∂ε ∂x M[n] (x ; ε),
0
we have n # # # # # # # # # # # # #∂ε M[n] (0; ε)# ≥ #∂ε M[n0 ] (0; ε)# − #∂ε j (0; ε)M [j ] (0; ε) # , j =n0 +1
and we can bound ∞ # # # # # # # # [n0 ] [n0 ] M (0; ε) ≥ M (0; ε) − #∂ε # #∂ε j0 #
j =j0 +1 T ∈S R j,n0
≥
|∂ε VT (0; ε)|
j0 j −1 # # # j0 j0 −1 ## |ε| |ε| 0 #Gj0 # , Gj0 # + O |ε|j0 −1 |ε| ≥ 2 4
(8.6)
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where we have reasoned as at the end of the proof of Lemma 8.1 in order to bound ∂ε VT (0; ε), and have used Lemma 6.4 and Remark 7.3 in order to fix p in (7.3). Hence # j # # # √ j0 j0 −1 0 # # |ε|j0 −1 #Gj0 # + O |ε|j0 −1 ε ≥ |ε| Gj0 ≡ ᒊ1 |ε|j0 −1 , #∂ε M[n] (0; ε)# ≥ 4 8 by the second inequality in (8.4) and by proceeding as at the end of the proof of Lemma 7.5 (see also Remark 7.6). Furthermore one has # # x # # # # [n] # ≤ |x| max ##∂ε ∂x M[n] (x; ε)## ≡ ᒊ2 |ε| |x| # M dx ∂ ∂ (x ; ε) (8.7) ε x # # x
0
because of Lemma 8.1, and the assertion is proved. Lemma 8.3. There are two positive constants b and ξ such that, for ε0 small enough and εm = 2−m ε0 , one has meas(E [∞] ) = meas(Em ∩ E∗ ) ≥
εm ξ , 1 − bεm 2
(8.8)
where meas denotes the Lebesgue measure. The constants b and ξ are independent of m. 2 Proof. Define I [n0 ] = ∅ and I [n] = E [n−1] \ E [n] for n ≥ n0 + 1; note that I := [n] = E \ E [∞] . Recall also that we have set E [n0 ] = E . ∪∞ m m n=n0 I For all n ≥ n0 + 1 and for all ν ∈ Zd∗ define # # ' & # # I [n] (ν) = ε ∈ E [n−1] : #iω · ν − M[n−1] (ω · ν; ε)# ≤ C1 |ν|−τ1 .
(8.9)
Each set I [n] (ν) has “center” in a point ε [n] (ν), defined implicitly by the equation iω ·ν − M[n] (ω · ν; ε[n] (ν)) = 0, where we are using the Whitney extension of M[n] (ω · ν; ε) outside E [n−1] . Therefore one has to exclude from the set E [n−1] all the values ε around ε [n] (ν) in [n] I (ν), and this has to be done for all ν ∈ Zd∗ satisfying |ω · ν| ≤
# 3 ## [n] # #M (ω · ν; ε)# , 2
(8.10)
because otherwise one can bound |iω · ν − M[n] (ω · ν; ε)| ≥ |ω · ν|/3 ≥ C1 |ν|−τ1 as soon as τ1 ≥ τ and C1 ≤ C0 /3. For ε small enough and for all n ≥ n0 one can bound |M[n] (0; ε) − M[n0 ] (0; ε)| ≤ n /τ n /τ 2D |ε|e−D 2 0 1 ≤ β1 B1 |ε|e−β2 B2 2 0 1 , by the third inequality in (7.5) of Lemma 7.5, applied repeatedly from scale n0 + 1 to scale n, and having used that β2 B2 < D and √ 0] j0 |ε|), if p in (7.3) 2D ≤ β1 B1 (see Remark 7.6), |M[n0 ] (0; ε)−M[n j0 (0; ε)| = O(|ε| [n0 ]
0] −B2 2 is chosen according to Remark 7.3, and |M[n j0 (0; ε) − Mj0 (0; ε)| ≤ B1 |ε|e by Lemma 6.3, so that one finds # # [n0 ] # [n] # #M (0; ε) − Mj0 (0; ε)# < 2Gj0 |ε|j0 ,
n0 /τ1
,
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if n0 is fixed as in Remark 6.6, so that 2β1 B1 |ε |e−β2 B2 2 0 1 ≤ |ε|j0 |Gj0 |. Therefore one can bound # # # # # # # [n] # # # # # #M (x; ε)# ≤ #M[n] (0; ε)# + #M[n] (x; ε) − M[n] (0; ε)# # # # [n0 ] # ≤ #Mj0 (0; ε)# + 2Gj0 |ε|j0 + D |ε| |x| # 3 ## # ≤ 3Gj0 |ε|j0 + D |ε| #M[n] (x; ε)# , 2 n /τ
with x = ω · ν, for all ν satisfying (8.10). We can conclude that there exists a constant j
ᑞ such that one has |M[n] (ω · ν; ε)| ≤ εm0 ᑞ for all ν satisfying (8.10).
Hence we have to consider only the vectors ν ∈ Zd∗ satisfying not only (8.10) but j also the inequality |ω · ν| < 2εm0 ᑞ, i.e. for all ν ∈ Zd∗ such that ! |ν| ≥
"1/τ
C0
:= N0 .
j
2εm0 ᑞ
(8.11)
We call N0 the set of ν ∈ Zd∗ which satisfy (8.10) and (8.11). For such ν, by setting x = ω · ν, one has # # 3 j # # #∂ε M[n] (x; ε)# ≥ ᒊ1 |ε|j0 −1 − ᒊ2 |ε| 2εm0 ᑞ 2 ᒊ1 j0 −1 ᒊ1 j −1 3ᒊ2 3/2 j0 ≥ j −1 εm 1− εm 2 ᑞ ≥ j εm0 , 20 2 ᒊ1 20
(8.12)
so that the measure of the corresponding excluded set, which can be written as
I [n] (ν)
dε =
1
−1
dt
dε(t) , dt
(8.13)
with ε(t) defined by iω · ν − M[n] (ω · ν; ε(t)) = tC1 |ν|−τ1 , will be bounded by
I [n] (ν)
dε ≤
1
−1
dt C1 |ν|−τ1
1 |∂ε
M[n] (ω
· ν; ε(t))|
≤
2j0 +1 j −1 ᒊ1 εm0
C1 , |ν|τ1
(8.14)
by (8.12). This yields that we have to exclude from E [n−1] a set I [n] = ∪ν∈N0 I [n] (ν) of measure bounded by meas(I [n] ) ≤
meas(I [n] (ν)) ≤ const.
ν∈N0
≤ const.
C1 j −1 εm0
!
j
εm0 C1
"(τ1 −d)/τ
C1 |ν|−τ1 j0 −1 ε m |ν|≥N0
1+ξ = const.εm ,
(8.15)
where ξ = j0 (τ1 − τ − d)/τ , so that ξ > 0 if τ1 > τ + d, which fixes the value of τ1 .
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We can easily prove that there exist two positive constants E1 and E2 such that one has # √ # n/τ1 # # [n] (8.16) #ε (ν) − ε [n−1] (ν)# ≤ εm E1 e−E2 2 for all n ≥ n0 + 1 and for all ν ∈ Zd∗ . By setting δε = ε[n] (ν) − ε [n−1] (ν) and x = ω · ν, we obtain (again by using Whitney extensions) 0 = ix − M[n] (x; ε[n] (ν)) = ix − M[n−1] (x; ε[n−1] (ν) + δε) − M[n] (x; ε[n] (ν)) + M[n−1] (x; ε[n] (ν)) = −∂ε M[n−1] (x; ε[n−1] (ν)) δε + o(δε) − M[n] (x; ε[n] (ν))−M[n−1] (x; ε[n] (ν)) , by (8.3) in Lemma 8.1; hence one can use that # # n/τ1 # # [n] #M (ω · ν; ε) − M[n−1] (ω · ν; ε)# ≤ D |ε| e−D 2 D n/τ1 ≤ |ε| β1 B1 e−β2 B2 2 β1 B1 ≤ B |ε|j0 e−β2 B2 2
n/τ1
,
(8.17)
with B a suitable constant, by the third inequality of (7.5) in Lemma 7.5, by (6.7) and by Remark 7.6. Hence by (8.12) and (8.17) we obtain (8.16) with E1 = 4B /ᒊ1 and E2 = β2 B2 . For all |ν| ≥ N0 fix n∗ = n∗ (ν) such that |ε [n∗ +1] (ν) − ε [n∗ ] (ν)| ≤ C1 |ν|−τ1 . One can choose n∗ (ν) ≤ const. τ1 log log |ν|. Then for all n0 + 1 ≤ n ≤ n∗ define J [n] (ν) as # # & ' # # J [n] (ν) = ε ∈ E [n−1] : #iω · ν − M[n−1] (ω · ν; ε)# < 2C1 |ν|−τ1 ; (8.18) by construction all the sets I [n] (ν) fall inside J [n∗ ] (ν) as soon as n > n∗ . Then we can bound meas(I) by the sum of the measures of the sets J [n0 +1] (ν), . . . , J [n∗ ] (ν) for all ν ∈ Zd∗ such that |ν| ≥ N0 . Such a measure will be bounded by const.
n∗ (ν)
|ν|≥N0
C1 |ν|−τ1 j0 −1 εm
1+ξ ≤ const.εm ,
(8.19)
with a value ξ smaller than ξ in order to take into account the logarithmic corrections due to the factor n∗ (ν). Proposition 8.4. Define the set of admissible values of E∗ as in Definition 5.11 with C1 = C0 /3 and τ1 > τ + d. Then one has meas(Em ∩ E∗ ) = 1. m→∞ meas(Em ) lim
2 Proof. It is an immediate consequence of the definitions and of Lemma 8.2.
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9. Properties of the Renormalized Expansion To complete the proof of existence of a quasi-periodic solution of (2.8) we have to show that the function defined by the renormalized expansion (5.13) solves Eq. (2.8). Set E+ = ∪∞ m=0 Em ∩ E∗ : such a set contains the admissible values of ε in [0, ε0 ]. Define analogously E− for the interval [−ε0 , 0], and set E = E+ ∪ E− . Lemma 9.1. For all ε ∈ E the function u(t) defined through (5.13) solves the equation (9.1) u = g R + εQu2 , where g is the pseudo-differential operator with kernel g(ω · ν) = 1/ iω · ν.
2
Proof. As in Sect. 8 of [14]. So far we proved that there exists a function u(t) = U (ωt; ε) which solves (2.8) for ε in a suitable large measure Cantor set E. For g given by g(t) = iεQ(t)u(t), Proposition 2.3 proves that φ(t) given in (2.6) solves (1.1) and is quasi-periodic. In principle, if we set ε = 0 + g , φ could be of the form ˜ 1 t, ω0 t, 0 t), φ(t) = (ω1 t, ω0 t, 0 t, ε t) ≡ eiε t (ω as it depends on u(t), and an extra frequency arises from the integral of the average of ˜ is of the g0 + g in the definition of (t). But this is not the case, because the function ˜ = (ω1 t, ω0 t), that is its dependence on t is only through the variables ω0 t and form ω1 t. This follows from the following property. Lemma 9.2. Let u be the function defined through the renormalized expansion (5.13): 2 then u[k] ν = 0 requires that in ν = (m, n1 , n2 ) one has n2 = 2. Proof. The proof is by induction on k. For k = 0 the result is obvious from the relation (0) (1) (2) (iω · ν)uν = Rν in (3.7) and from the identity Rν = Pm Fn1 δn2 ,2 . Let us assume that
] u[k ν ∝ δn2 ,2 for all k < k. Then to order k the second relation in (3.7) yields that one can have u[k] ν = 0 only if ν = ν 0 + ν 1 + ν 2 : for the last component n2 of the vector ν the (−2) identity Qν = δm,0 Fn1 δn2 ,−2 and the inductive assumption give n2 = −2+2+2 = 2.
Hence u(t) = e2i0 t U˜ (ω1 t, ω0 t), with U˜ analytic and periodic in its arguments. By taking into account that one has Q(t) = e−2i0 t−2iψ0 (t) , with ψ0 (t) depending on t only through the variable ω0 t, one has Q(t)u(t) = e−2iψ0 (t) U˜ (ω1 t, ω0 t). As a consequence φ(t) is a quasi-periodic function with d fundamental frequencies ω1 , ω0 , ε , and the dependence on the last frequency is only through the factor eiε t , exactly as in the unperturbed case (2.2). As anticipated in Remark 2.2 the same result can be obtained by starting from the unperturbed solution given by the second function in (2.4), and an analogous result is found, so that we can conclude that the system is reducible for ε ∈ E. So the solution u(t) describes the motion on a d-dimensional maximal torus which is the continuation in ε of an unperturbed d-dimensional torus. The rotation vector of the latter is ω = (ω1 , ω0 , 0 ), while, as an effect of the perturbation, only the last component of the rotation vector is changed into a new frequency ε = 0 + g : this provides a simple physical interpretation of the quantity g . It is likely that the new frequency ε is such that the vector (ω1 , ω0 , ε ) is still Diophantine. This does not follow directly from our analysis, but we expect that this is the case.
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10. Null Renormalization We are left with the case in which one has Gj = 0 for all j ∈ N. In such a case we need no resummations, as it will become clear from the analysis. Hence we use the simpler multiscale decomposition of the propagators given by (5.1), with C1 = C0 . The following result holds. Lemma 10.1. One has ψn−1 (x)ψn (x) = ψn−1 (x) and ψ0 (x) +
n
χj −1 (x)ψj (x) = ψn (x),
(10.1)
j =1
for all n ∈ N and for all x ∈ R. Proof. Both relations follow immediately from the definitions. Then we consider the same tree expansion leading to (5.2), where no resummation is performed. The following result allows us to get rid of some trees. Lemma 10.2. Suppose that one has Gj = 0 for all j ∈ N. Then in the tree expansion of (k) uν in (5.2) the sum over k,ν can be restricted only to trees which do not contain any vertex v such that one of the entering lines carries the same momentum of the exiting line. 2 Proof. If there were no scale labels this would follow from item (a) in Proposition 4.15. The presence of the scales could destroy in principle the compensation mechanism responsible for the cancellation among the values of the various trees. But it is sufficient (k) to note that the coefficient uν is obtained by summing over all the possible scale labels, and in this way we reconstruct for each line the original propagator 1/ iω · ν (just use (10.1) for n → ∞), hence we can apply the cited result. Remark 10.3. If Gj = 0 for all j ∈ N formal solubility of Eq. (1.4) requires no condition on the coefficients α (k) , which therefore can be arbitrarily fixed (cf. Lemma 4.5). For simplicity we can still fix α (k) = 0 for all k, even if this is not strictly necessary. Of course one can ask what happens for other choices of the coefficients α (k) , but we do not investigate further such a problem because the case in which all Gj are vanishing is rather special, and likely it can really arise only in trivial situations (like p1 ≡ 0). We define the clusters according to the definition previously done, whereas we slightly change the definition of the self-energy graph, to make it more suitable for our purposes in the present case (cf. [3]). An important feature is that the propagators are not changed by any resummation procedure, so that for any line the (two) scales for which the corresponding propagator is not vanishing are uniquely fixed by ν . Definition 10.4 (Self-Energy Graph). We call a self-energy graph any cluster T of a tree θ which satisfies out 1. T has only one entering line in T and only one exiting line T ; 2. The momentum of T is zero, i.e. ν T = v∈B(T ) ν v = 0.
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3. The mode labels ν v , v ∈ B(T ), satisfy the relation v∈B(T ) |ν v | < 2(next −4)/τ , where next is the minimum between the scales of the external lines of T . We call a self-energy line any line out T which exits from a self-energy graph T . We call a normal line any line which is not a self-energy line. The self-energy value is then defined as before (see (5.8)), with the only difference (n ) that now the propagators are g (because they are not renormalized). The aim of the last item in the definition of self-energy graph is that, given a selfenergy graph, if we sum over all the scales of the internal lines compatible with the cluster structure, which yields that for each line ∈ L(T ) one has n < next , if next = min{n out , n in }, then we reconstruct for each line a propagator ψnext +1 (ω · ν )/ iω · ν , T T with ψnext +1 (ω · ν ) = 1. The last assertion is implied from the following result. Lemma 10.5. For any self-energy graph T , by setting next = min{n out , n in }, one can T T have VT (ω · ν) = 0 only if n ≤ next − 2 for any line ∈ L(T ). 2 Proof. By definition of scales one has C0 2−next −1 ≤ |ω · ν| ≤ C0 2−next +1 . The third item in the definition of self-energy graph gives |ω · ν 0 | > C0 2−(next −4)/τ (see (5.9) for the definition of ν 0 ), hence by the Diophantine condition (3.5) on ω one obtains |ω · ν | ≥ |ω · ν 0 | − |ω · ν| ≥ C0 2−(next −4) − C0 2−(next −1) ≥ C0 2−(next −3) , so that χn −1 (ω · ν ) = 0 for n > next − 2. Definition 10.6 (Localization). For any self-energy graph T we can define the localized part of the self-energy value VT (ω · ν) as LVT (ω · ν) = VT (0),
(10.2)
and the regularized part as
1
RVT (ω · ν) = (ω · ν)
dt ∂VT (tω · ν),
(10.3)
0
where ∂ denotes derivative with respect to the argument, so that ∂VT (tω · ν) = ∂VT (x)/∂x|x=tω·ν . We shall call L and R the localization and regularization operator, respectively. By definition of self-energy value, one has ∂VT (tω · ν) = ε kT Fv v∈B(T )
∈L(T )
∂g (n ) (ω · ν (t))
g (n ) (ω · ν (t)) ,
(10.4)
∈L(T )\
where ν (t) = ν 0 if is not along the path connecting the external lines of T , and ν (t) = ν 0 + tν otherwise.
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The definition above suggests a further splitting of the tree values. With each selfenergy graph T we associate a localization label which can be either L or R: the first one means that we have to compute the self-energy value for ω · ν = 0, while the second one tells us that we have to replace VT (ω · ν) with RVT (ω · ν) as given by (10.3). Since a self-energy graph can contain other self-energy graphs, the application of the localization and regularization operators has to be performed iteratively by starting from the outermost (or maximal) self-energy graphs to end up with the innermost ones. Lemma 10.7. Suppose that one has Gj = 0 for all j ∈ N. Then in the tree expansion (k) 2 of uν in (5.2) only trees with localization label R have to been retained. Proof. Given a maximal self-energy graph T consider the localized part of its self-energy graph. First of all note that the entering line of T cannot enter the same vertex v which the exiting line of T comes out from (as a consequence of Lemma 10.2). For the remaining trees we can sum over all the scale labels compatible with the cluster structure, and apply Lemma 10.5 (which allows us to replace the support compact functions with 1). Then we can apply the cancellation mechanism leading to Lemma 4.12: indeed one immediately realizes that the cancellation works for fixed mode labels (see Remark 4.11). Then VT (0) = 0, so that we can replace VT (ω · ν) with RVT (ω · ν), as given by (10.4). Here ν is the momentum of the line entering T . Next look at a self-energy graph T contained inside T and which is maximal (that is the only self-energy graph containing T is T itself), and suppose we are considering a contribution to RVT (ω·ν) in which the derivative acts on some propagator external to T . 0 The momentum ν (t) flowing through the entering line in T of T is either ν (t) = ν in T
or ν (t) = ν 0 in + tν, so that for each line ∈ L(T ) one has either ν = ν 0 or T
ν = ν 0 + ν (t). So when we compute the localized part of the self-energy value of T , we have to put ν (t) = 0, and we can reason exactly as before for T : then the same cancellation mechanism applies. If instead the derivative in (10.4) acts on the self-energy value VT (ω · ν (t)) then we can write VT (ω · ν (t)) = LVT (ω · ν (t)) + RVT (ω · ν (t)), and of course the first term gives no contribution as it is a constant. Hence also in such a case we can get rid of the localized part of the self-energy value. We can iterate the argument until no further self-energy graph is left, and the assertion follows. Hence we have to consider the tree expansion (5.2), and retain only self-energy clusters with localization label R. The discussion then becomes standard (see for instance [16]), and for each self-energy graph T , if ν is the momentum flowing through its external lines, we obtain a gain factor ω · ν, which compensate exactly one of the propagators of the external lines of T , say that of the exiting line (self-energy line). Of course one has to control that no line is differentiated more than once, but this is a standard argument (again we refer to [16] for details). At the end we obtain that Val(θ ) admits a bound like (5.5), with the only difference that the propagators can be differentiated so that they have to be bounded as if they were quadratic and not linear. On the other hand only normal lines have to be considered, as the self-energy lines are compensated by the mechanism described above, and they are bounded through (5.4). And the bound (5.4) still holds with the new definition of self-energy graph, as shown in [16].
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The conclusion is that the series defining u(t) is convergent, and it turns out to be analytic in ε. In particular this means that no value of ε has to be discarded in such a case. Moreover g = 0, because g = iε Qu , and Qu = 0 by item (a) in Proposition 4.15 and the hypothesis that one has Gj = 0 for all j ∈ N. In particular one has ε = 0 . Therefore the case in which Gj = 0 for all j corresponds to having an integrable system. Note that the condition Gj = 0 for all j ∈ N is a condition on the perturbation itself, so that it is not something that has to be checked while carrying on any iterative scheme to solve the problem.
References 1. Avron, J.E., Simon, B.: Almost periodic Hill’s equation and the rings of Saturn. Phys. Rev. Lett. 46(17), 1166–1168 (1981) 2. Barata, J.C.A.: On Formal Quasi-Periodic Solutions of the Schr¨odinger Equation for a Two-Level System with a Hamiltonian Depending Quasi-Periodically on Time. Rev. Math. Phys. 12(1), 25–64 (2000) 3. Bartuccelli, M.V., Gentile, G.: Lindstedt series for perturbations of isochronous systems: a review of the general theory. Rev. Math. Phys. 14(2), 121–171 (2002) 4. Bohr, H.: Ueber fastperiodische ebene Bewegungen. Comment. Math. Helv. 4, 51–64 (1932) Bohr, H.: Kleinere Beitr¨age zur Theorie der Fastperiodischen Funktionen. Danske Vid. Selsk. Mat.-Fys. Medd. 10(10), 5–11 (1930) 5. Broer, H., Puig, J., Sim´o, C.: Resonance tongues and instability pockets in the quasi-periodic HillSchr¨odinger equation. Commun. Math. Phys. 241(2–3), 467–503 (2003) 6. Broer, H., Sim´o, C.: Hill’s equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena. Bol. Soc. Bras. Mat. 29(2), 253–293 (1998) 7. Cheng, Ch.-Q.: Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltonian systems. Commun. Math. Phys. 177(3), 529–559 (1996) 8. Cheng, Ch.-Q.: Lower-dimensional invariant tori in the regions of instability for nearly integrable Hamiltonian systems. Commun. Math. Phys. 203(2), 385–419 (1999) 9. Chierchia, L.: Quasi-periodic Schr¨odinger operators in one dimension, absolutely continuous spectra, Bloch waves and integrable Hamiltonian systems. Firenze: Quaderni del Consiglio Nazionale delle Ricerche, 1986 10. Chierchia, L.: Absolutely continuous spectra of quasi-periodic Schr¨odinger operators. J. Math. Phys. 28(12), 2891–2898 (1987) 11. Dinaburg, E.I., Sina˘ı, Ja.G.: The one-dimensional Schr¨odinger equation with quasiperiodic potential. Funk. Anal. i Pril. 9(4), 8–21 (1975) 12. Eliasson, L.H.: Floquet solutions for the 1-dimensional quasi-periodic Schr¨odinger equation. Commun. Math. Phys. 146(3), 447–482 (1992) 13. Gallavotti, G., Gentile, G.: Hyperbolic low-dimensional invariant tori and summations of divergent series. Commun. Math. Phys. 227(3), 421–460 (2002) 14. Gentile, G.: Quasi-periodic solutions for two-level systems. Commun. Math. Phys. 242(1–2), 221– 250 (2002) 15. Gentile, G., Gallavotti, G.: Degenerate elliptic resonances. Commun. Math. Phys. 257(2), 319–362 (2005) 16. Gentile, G., Mastropietro, V.: Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications. Rev. Math. Phys. 8(3), 393–444 (1996) 17. Herman, M.: In: Some open problems in dynamical systems. In: Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. Extra Vol. II, 797–808 (1998) 18. Hinton, D.B., Shaw, J.K.: On the absolutely continuous spectrum of the perturbed Hill’s equation. Proc. London Math. Soc. 50(3), no. 1, 175–192 (1985) 19. Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84(3), 403–438 (1982) 20. Katznelson, Y.: An Introduction to harmonic analysis, Dover, New York, 1978 21. Krikorian, R.: R´eductibilit´e presque partout des flots fibr´es quasi-p´eriodiques a` valeurs dans des ´ groupes compacts. Ann. Sci. Ecole Norm. Sup. (4) 32(2), 187–240 (1999) 22. Krikorian, R.: R`eductibilit´e des syst`emes produits-crois´es a` valeurs dans des groupes compacts. Ast´erisque 259, vi+216 pp (1999)
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23. Krikorian, R.: Global density of reducible quasi-periodic cocycles on T 1 × SU(2). Ann. of Math. (2) 154(2), 269–326 (2001) 24. Magnus, W., Winkler, S.: Hill’s Equation. New York, Dover, 1979 25. Moser, J., P¨oschel, J.: An extension of a result by Dinaburg and Sina˘ı on quasiperiodic potentials. Comment. Math. Helv. 59(1), 39–85 (1984) 26. Rofe-Beketov, F.S.: A finiteness test for the number of discrete levels which can be introduced into the gaps of the continuous spectrum by perturbations of a periodic potential (in Russian). Dokl. Akad. Nauk SSSR 156, 515–518 (1964) 27. R¨ussmann, H.: On the one-dimensional Schr¨odinger equation with a quasiperiodic potential. In: Nonlinear dynamics (Internat. Conf., New York, 1979), Ann. New York Acad. Sci. 357, New York: New York Acad. Sci., 1980 28. R¨ussmann, H.: Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition. Ergodic Theory Dynam. Systems 22(5), 1551–1573 (2002) 29. Sorets, E., Spencer, T.: Positive Lyapunov exponents for Schr¨odinger operators with quasi-periodic potentials. Commun. Math. Phys. 142(3), 543–566 (1991) 30. Szebehely, V.G.: Theory of orbits. New York: Academic Press, 1967 ˇ 31. Zeludev, V.A.: The eigenvalues of a perturbed Schr¨odinger operator with periodic potential (in Russian). Problems of Mathematical Physics, no. 2, Spectral Theory, Diffraction Problems, Leningrad: Izdat. Leningrad. Univ., 1967, pp. 108–123 ˇ 32. Zeludev, V.A.: The perturbation of the spectrum of the one-dimensional selfadjoint Schr¨odinger operator with periodic potential (in Russian), Problems of Mathematical Physics, no. 4, Spectral Theory. Wave Processes, Leningrad: Izdat. Leningrad. Univ., 1970, pp. 61–82 Communicated by G. Gallavotti
Commun. Math. Phys. 260, 445–454 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1414-6
Communications in
Mathematical Physics
Free Energy as a Geometric Invariant Mark Pollicott1 , Howard Weiss2 1 2
Department of Mathematics, The University of Manchester, Oxford Road, M13 9PL, Manchester, England. E-mail: [email protected] Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA. E-mail: [email protected]
Received: 1 November 2004 / Accepted: 6 March 2005 Published online: 31 August 2005 – © Springer-Verlag 2005
Abstract: The free energy plays a fundamental role in statistical and condensed matter physics. A related notion of free energy plays an important role in the study of hyperbolic dynamical systems. In this paper we introduce and study a natural notion of free energy for surfaces with variable negative curvature. This geometric free energy encodes a new type of marked length spectrum of closed geodesics, which lies between the well-known marked length spectrum (marked by the corresponding element of the fundamental group) and the unmarked length spectrum. We prove that the free energy parametrizes the boundary of the domain of convergence of a Poincar´e series which also encodes this spectrum. We also show that this new length spectrum, or equivalently the geometric free energy, is not an isometry invariant. In the final section we use tools from multifractal analysis to effect a fine asymptotic comparison of word length and geodesic length of closed geodesics. We hope that our geometric understanding of free energy will provide new insight into this fundamental physical and dynamical quantity. 0. Introduction The free energy, which can be viewed as a generating function for the sequence of energy levels of a system, plays a fundamental role in statistical and condensed matter physics. A related notion of free energy plays an important role in hyperbolic dynamical systems, where it can be viewed as a generating function for the orbital averages of a function over the closed orbits. In this paper we introduce and study a natural notion of free energy for surfaces with variable negative curvature. The geometric free energy is defined as a generating function for the lengths of closed geodesics indexed by the lengths of corresponding words in the fundamental group, as The work of the second author was partially supported by a National Science Foundation grant DMS-0355180. This work was completed during a visit by the first author to Penn State as a Shapiro Fellow.
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follows. Let V be a compact negatively curved surface and let denote a set of generators for the fundamental group π1 (V ) of V . The symbol γ will always denote a closed geodesic, [γ ] the free homotopy class of γ in π1 (V ), and l(γ ) the geodesic length of γ . Every free homotopy class contains a unique closed geodesic. We let |γ | denote the word length of the free homotopy class of [γ ], i.e., where |γ | is the smallest number of generators from needed to represent an element in [γ ]. We define the free energy of V by 1 F (t) = F,V (t) lim exp (−tl(γ )). log n→+∞ n [γ ]∈ |γ |=n
This definition arises as a special case of the dynamical free energy for the geodesic flow, where the geodesic flow is coded by the geometric Bowen-Series Markov partition. This is carefully explained in Sect. 1. We also prove that the free energy parametrizes the boundary of the domain of convergence of the related Poincar´e series exp (−al(γ ) − b|γ |). [γ ]∈
There is extensive literature on the rigidity properties of the marked and unmarked length spectrum of closed geodesics on a negatively curved surface. The marked length spectrum consists of the lengths of closed geodesics marked by the corresponding free homotopy class of the geodesic, i.e., the sequence of pairs {([γ ], l(γ )), γ ∈ π1 (V )}. The marked length spectrum is an isometry invariant [Cro, Ota]. The unmarked length spectrum is the sequence consisting of just the lengths of the closed geodesics and is known not to be an isometry invariant [Bus, McK, Sun, Vig, Wol]. Our geometric free energy encodes a third type of marked length spectrum of closed geodesics given by the sequence of pairs {(|γ |, l(γ )), γ ∈ π1 (V )}, which lies between the marked length spectrum and the unmarked length spectra. The relationship between the word length and geometric length of closed geodesics was first studied by Milnor [Mil]. We show that this length spectrum, or equivalently the geometric free energy, is not an isometry invariant. In the final section we use tools from multifractal analysis to effect a fine asymptotic comparison of word length and geodesic length of closed geodesics. We hope that our geometric understanding of free energy will provide new insight into this fundamental physical and dynamical quantity. This note can be viewed as a continuation of [PW], where the authors study the free energy of a classical one-dimensional lattice spin system as a dynamical invariant. All of the facts from thermodynamic formalism we require can be found in [Rue] and [PP]. 1. Free Energy and Poincar´e Series Let φ t T1 V → T1 V denote the geodesic flow on the unit tangent bundle of V . Our approach starts with the Bowen-Series coding [BS, BKS] to obtain a symbolic representation of the geodesic flow on T1 V . While there are other methods of constructing Markov partitions for the geodesic flow [Bow, Rat, Sin], this method, which codes intersections of geodesics with sides of a fundamental polygon, is the only known coding scheme which faithfully codes the lengths and combinatorics of (essentially) all closed geodesics. It produces a suspension flow over a subshift of finite type with a roof function whose Birkhoff sum over a periodic orbit of period n is precisely the length of the
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corresponding closed geodesic, where n is also the word length in π1 (V ) of the closed geodesic. A very detailed account of this approach appears in the article of Adler and Flatto [AF], to which we refer the reader for more details. The Bowen-Series coding starts with a specially chosen fundamental polygon for the fundamental group of V in the universal covering surface [AF, Appendix I]. In particular, each side of the finite-sided polygon is a subarc of a closed geodesic. One then identifies various pairs of the geodesic subarcs using covering transformations corresponding to some element of [AF, Theorem 3.1]. One then verifies that the restrictions of these identification maps to the boundary of the universal covering (which can be identified with the unit circle) determines a map T of the unit circle which is Markov [AF, Theorem 3.4 and Theorem 6.1] and eventually expanding [AF, Theorem 6.3]. By construction, T is orbit equivalent to the action of on the boundary of the universal covering. In order to study particular properties of the geodesic flow, it is usually necessary to code pairs of endpoints of appropriate geodesics using two dimensional Markov Partitions [AF, pp. 321–328]. However, for periodic points it suffices to consider only the + + map T . The Markov map T is naturally coded by a subshift of finite type σ : A → A . + There is also a natural H¨older continuous roof function r : A → R whose interpretation is as the time required for points on the special geodesic arcs to intersect other special geodesic arcs under the geodesic flow. (In practise, it is usually convenient to take + r(x) = log |T (π(x))|, where π is the semi-conjugacy map from A to the circle, which + only differs by a coboundary). One then defines the suspension space Xr = A × R/Z, where Z is the group of maps generated by (x, y) → (σ x, y − r(x)), and the suspension flow σrt Xr → Xr induced by the maps (x, t) → (x, y + t) [PP]. The geodesic flow φ t and the suspension flow are related by the existence of a bounded-to-one continuous surjection p Xr → T1 V such that φ t ◦ p = p ◦ σrt . Moreover, given two negatively curved metrics on the same topological surface, + + the underlying subshifts of finite type σ : A → A from the Bowen-Series coding of their geodesic flows are the same. Finally, every closed geodesic γ corresponds + to a periodic orbit {x, σ x, . . . , σ n−1 x} of A with σ n x = x, where the Birkhoff sum n−1 Sn r(x) = k=0 r(σ k x) = l(γ ) and n = |γ |. This correspondence is one-to-one, except for an at most finite set of exceptional prime closed geodesics, which have no effect on our results. In particular, the exceptional closed geodesics are these corresponding to end points of intervals in the Markov partition for T cf. [AF, p.319]. These special properties of the Bowen-Series coding are crucial for our analysis. If one could construct a symbolic coding of the geodesic flow for any Riemannian metric satisfying these properties, then all the results in this paper would immediately carry through to this metric. In [AR] the authors state technical conditions for a more general Markov partition to possess these properties, but there are very few examples in dimensions greater than two which have been shown to satisfy these properties. Using the Bowen-Series coding, we define the free energy of the negatively curved + surface V to be the free energy of the roof function r A → R, i.e., F (t) = F,V (t) P (−tr), where P denotes the thermodynamic pressure defined on all real valued continuous + functions on A . Using the periodic orbit definition of pressure [PP] we have 1 log exp (−tSn r(x)). n→+∞ n n
P (−tr) = lim
σ x=x
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Asymptotic slope −B
a
h
1 b
Asymptotic slope −A
Fig. 1. The Curve L
It immediately follows from faithful coding properties of closed geodesics for the Bowen-Series coding that the free energy of V has the following geometric realization, which we define to be the free energy of V : 1 log n→+∞ n
F (t) = F,V (t) lim
exp (−tl(γ )).
[γ ]∈π1 (V ) |γ |=n
It follows from standard facts in thermodynamic formalism [PP, Rue] that this free energy is a smooth and strictly convex function. We now define a modified Poincar´e series for V and by ρ(a, b) = ρ,V (a, b) exp (−al(γ ) − b|γ |). [γ ]∈π1 (V )
Our next goal is to show that the free energy parametrizes the boundary curve separating the domain of convergence of this Poincar´e series from the domain of divergence. We recall that by a classical result of Milnor [Mil], there exist A, B > 0 such that A ≤ l(γ )/|γ | ≤ B for all closed geodesics γ . If we choose A = inf γ l(γ )/|γ | and B = supγ l(γ )/|γ | then using the Anosov closing lemma, it is easy to show that the ratios {l(γ )/|γ | : γ = closed geodesic} are dense in the interval [A, B]. It follows from Milnor’s result that the Poincar´e series ρ(a, b) converges provided a, b > 0 are sufficiently large. We denote by R = R(, V ) = {(a, b) ∈ R2 : ρ(a, b) < +∞} the domain of convergence of the Poincar´e series and L = L(, V ) the boundary curve of R (see Fig. 1). The following proposition provides the important link between the free energy and the Poincar´e series. Proposition 1. The free energy F parametrizes the boundary curve L in the sense that L = {(a, F (a)} for −∞ < a < +∞.
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Proof. Using the key property of the Bowen-Series coding we can rewrite the Poincar´e series in the following form: ρ(a, b) = ρ,V (a, b)
∞ 1 exp (−aSn r(x) − bn). n n n=1
σ x=x
By the root test, this infinite series converges if 1/n exp (P (−ar − b)) = lim exp (−aSn r(x) − bn) < 1. n→+∞
σ n x=x
In particular, we immediately see that the sets L and R have the following simple interpretation in terms of symbolic dynamics R = {(a, b) ∈ R2 : P (−ar − b) < 0} and L = {(a, b) ∈ R2 : P (−ar − b) = 0}. Since the pressure P (−ar −b) = 0 for (a, b) ∈ L, we see that b = P (−ar) = F (a).
We now define the generating function for word length in π1 (V ) by G(z) =
∞
zn card{γ : |γ | = n}.
n=1
This is well known to be a rational function [Can, Eps]. The next proposition states that L is a smooth curve and identifies some special values on L. Proposition 2. (a) The curve L is real analytic and strictly convex. (b) The points (0, 1) and (h, 0) lie on L, where 1 log Card{γ : |γ | = n}, n→+∞ n
h = lim
i.e., exp(−h) is the radius of convergence of the generating function G(z) for word length in π1 (V ). (c) The asymptotic slope of L as a → ±∞ is −A and −B, respectively. Proof. Parts (a) and (b) follow easily from standard properties of pressure [PP]. In + + particular, h is the topological entropy of the subshift σ : A → A . For part (c) we recall that the slope of the curve L at (a, F (a)) is F (a) = − + rdµ−ar , where µ−ar is the Gibbs measure for −ar [PP]1 . By the variational A
+ A σ −invariant probability measure µ on A is called Gibbs if there exists a H¨older continuous + function g A → R such that the pressure P (g) satisfies 1
P (g) =
sup
+ µ∈M(A )
hµ (σ ) +
+ A
gdµ ,
where the supremum is taken over the simplex M of all σ −invariant Borel probability measures µ on + . A
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principle [PP] we have that hµ−ar (σ ) − a
+ A
rdµ−ar ≥ hµ (σ ) − a
for all σ -invariant probability measures µ. Thus
+ A
+ A
rdµ,
rdµ−ar ≤ inf µ
+ A
rdµ + 2h/a,
+ + → A . In particular, where h denotes the topological entropy of the subshift σ A letting a → +∞, and using the weak density of probability measures supported on single periodic orbits for the subshift, we see that rdµ−ar = − inf rdµ = − inf l(γ )/|γ | = −A. lim F (a) = − lim
a→+∞
a→+∞ + A
µ
The proof of the second part of (c) is similar.
+ A
[γ ]∈π1 (V )
2. Free Energy as an Isometry Invariant Let V1 and V2 be negatively curved surfaces and let be a symmetric generating set for π1 (V1 ). If φ V1 → V2 is an isometry, then clearly the free energy of V1 with respect to coincides with the free energy of V2 with respect to the pushed forward generating set φ∗ , i.e., F,V1 = Fφ∗ ,V2 . Now suppose that V1 and V2 are negatively curved surfaces with generating sets 1 of π1 (V1 ) and 2 of π1 (V2 ), and assume that the free energies coincide, i.e., F1 ,V1 = F2 ,V2 . A natural rigidity question is whether this implies that the surfaces V1 and V2 are isometric? In this section we show that this is, in general, false. Thus in this sense, free energy is not a complete invariant of isometry. Proposition 3. There exist two negatively curved hyperbolic surfaces V1 and V2 with generating sets 1 of π1 (V1 ) and 2 of π1 (V2 ) such that F1 ,V1 = F2 ,V2 , but V1 and V2 are nonisometric. Proof. We verify that one of the standard constructions of isospectral but nonisometric hyperbolic surfaces [Bus, p. 304], with a suitable choice of generators for the fundamental group, provides a pair of nonisometric hyperbolic surfaces having the free energy. We follow Buser’s description almost verbatim. The building blocks for the construction are hyperbolic surfaces B of signature (0, 5) with boundary geodesics α1 , α2 , β, γ1 and γ2 , satisfying (α1 ) = (α2 ) < (β) < (γ1 ) = (γ2 ) < (η1 ) < (η2 ) ≤ 1. See Fig. 2. It follows from the collar lemma in hyperbolic geometry that the only isometry of such a five holed sphere is the identity. The two interior closed geodesics η1 and η2 decompose the (0, 5) surface into three (0, 3) surfaces (pairs of pants), which have been pasted together with a one-quarter twist. We now glue together eight copies (building blocks) B0 , . . . , B7 of B with twist parameter zero according to the identifications shown in Fig. 3 to obtain two compact hyperbolic surfaces V1 and V2 . Buser proves that V1 and V2 are isospectral but not isomorphic. The proof that V1 and V2 are not isomorphic is a consequence of the above mentioned collar lemma. We now construct a generating set for the fundamental group of V1 and V2 . For, say V1 , choose all the boundary closed geodesics α1 , α2 , β, γ1 and γ2 , as well as all the interior closed geodesics η1 and η2 , for each of the eight building blocks. Several pairs of boundary closed geodesics are glued together to form V1 and V2 , and it is easy to see
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Fig. 2. The Building Block (from [Bus])
Fig. 3. The Surfaces V1 and V2 (from [Bus])
that of the 8 × 5 = 40 boundary closed geodesics on the union of the eight building blocks, only 20 of these are distinct closed geodesics on V1 and V2 . Since V1 and V2 have genus 13, these 20 closed geodesics do not form a generating set. Since none of the interior closed geodesics are identified, all 8 × 2 = 16 are distinct closed geodesics on V1 and V2 . We let denote the generating set of the fundamental group consisting of these 20 + 16 = 36 generators. We claim that the free energies of V1 and V2 coincide with respect to the generating set , i.e., F,V1 = F,V2 .
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To see this, we exploit the very simple method of transplanting closed geodesics from M1 to M2 , due to Buser and B´erard, and which is clearly explained in Sect. 11.5 of [Bus]. This provides a one-to-one length-preserving correspondence between the closed geodesics on both surfaces. In short, Buser decomposes a closed geodesic c on V1 into a sequence of disjoint subarcs cj , with each subarc contained in one building block, and maps the interior of each geodesic subarc cj to the interior of a geodesic subarc cj∗ on V2 using the family of identity mappings φkk ∗ from the interior of Bk to the interior of Bk ∗ . Thus with the obvious interpretation, the transplantation mapping is the identity on the interiors of the eight building blocks. The very simple form of this correspondence between closed geodesics on the two surfaces is a consequence of the above mentioned fact that the only isometry of the building blocks is the identity. It follows that every closed geodesic on M1 , which is not an element of , and its transplanted closed geodesic on M2 , have the same length, and furthermore, have the same number of intersections with the closed geodesics in . This completes the proof. 3. The Rigidity of a New Type of Marked Length Spectrum For a hyperbolic or negatively curved surface V , one defines the unmarked length spectrum of V to be the set of lengths of all closed geodesics, i.e., LV = {l(γ ), γ ∈ π1 (V )}. The marked length spectrum of V consists of the lengths of closed geodesics, marked by the corresponding element of the fundamental group, i.e., the sequence MV = {(l(γ ), [γ ]), γ ∈ π1 (V )}. There is a significant amount of literature studying whether, or to what extent, the marked and unmarked length spectrum determine the metric [Bus, Cro, McK, Ota, Sun, Vig, Wol]. Motivated by our study of free energy, it is natural to study the length spectrum which is marked by the word length of the closed geodesic with respect to a fixed set of generators. This natural length spectrum lies between the well-known marked length spectrum and the unmarked length spectrum. More precisely, let V be a compact negatively curved surface and let denote a set of generators for the fundamental group π1 (V ) of V . We define the word length marked length spectrum of V : M,V = {(l(γ ), |γ |), [γ ] ∈ π1 (V )}. Let V1 and V2 be negatively curved surfaces and let be a generating set for π1 (V1 ). If φ : V1 → V2 is an isometry, then M,V1 = Mφ∗ ,V2 . Now suppose that V1 and V2 are negatively curved surfaces with generating sets 1 of π1 (V1 ) and 2 of π1 (V2 ), and assume that M1 ,V1 = M2 ,V2 . It immediately follows from definitions that F1 ,V1 = F2 ,V2 . A natural rigidity question is whether M1 ,V1 = M2 ,V2 implies that the surfaces V1 and V2 are isometric? The following proposition shows this is false. Proposition 4. There exist two negatively curved hyperbolic surfaces V1 and V2 with generating sets 1 of π1 (V1 ) and 2 of π1 (V2 ) such that MV1 ,1 = MV2 ,2 , but V1 and V2 are not isometric. Proof. The counterexample constructed in Sect. 2 also serves as a counterexample here. By construction, every closed geodesic on V1 and its transplanted mate on V2 have the same length and the same word length with respect to the generating sets 1 and 2 respectively.
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4. Asymptotic Comparison of Geodesic Length and Word Length For every unit tangent vector v one can consider the associated geodesic with c(0) ˙ = v. For any T > 0, the geodesic arc c([0, T ]) can be “closed up”, by adding a second piece of geodesic arc of bounded length, to give a closed geodesic which we denote γv,T . If the limit limT →+∞ l(γv,T )/|γv,T | exists, then it is easy to see that the limit α = α(v) is independent of the particular construction of γv,T . The Birkhoff ergodic theorem implies that for almost all v (with respect to the Liouville measure) the limit does exist and attains the constant value α0 , where A ≤ α0 ≤ B. For A ≤ α ≤ B we define the dimension spectrum for geodesic length and word length l(γv,T ) f (α) = dimH v ∈ T1 V : lim =α , T →+∞ |γv,T | where dimH denotes Hausdorff dimension [Fal]. The next result can be viewed as an asymptotic refinement of Milnor’s result. Proposition 5. (1) The dimension spectrum f (α) is an analytic function on (A, B). (2) For each α ∈ [A, B], the dimension spectrum f (α) ≤ 3, with equality if and only if α = α0 . (3) For each α ∈ [A, B], the level set {v : limT →+∞ l(γv,T )/|γv,T | = α} is uncountable, dense, and supports an invariant Gibbs measure. Proof. This is an immediate application of results on multifractal analysis in [Pes, PeW, Wei]. From the symbolic dynamics we can see that a unit tangent vector v, and its future + orbit, correspond to an infinite word x ∈ A . Moreover, l(γv,T )/|γv,T | converges to α if and only the Birkhoff average Sn r(x)/n converges to α. Since the roof function r is not cohomologous to a constant, the multifractal analysis for the Birkhoff sum gives results corresponding to (1), (2), and (3) at the symbolic level, and using the above mentioned correspondence, these immediately translate into the geometric results claimed in the proposition. References [AF]
Adler, R.L., Flatto, L.: Geodesic Flows, Interval Maps, and Symbolic Dynamics. Bull. Amer. Math. Soc. 25, 229–334 (1991) [AR] Anderson, J., Rocha, A.: Analyticity of Hausdorff dimension of limit sets of Kleinian Groups. Ann. Acad. Sci. Fenn. Math. 22, 349–364 (1997) [BKS] Bedford, T., Keane, M.S., Series, C.: Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces. Oxford: Oxford Science Publications 1997 [Bow] Bowen, R.: Symbolic Dynamics for Hyperbolic Flows. Amer. J. Math. 95, 429–460 (1973) [BS] Bowen, R., Series, C.: Markov Maps Associated with Fuchsian Groups. Inst. Hautes Etudes Sci. Publ. Math. 50, 153–170 (1979) [Bus] Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Basel-Boston: Birkh¨auser (1992) [Can] Cannon, J.: The Combinatorial Structure of Cocompact Discrete Hyperbolic Groups. Geom. Dedicata 16, 123–148 (1984) [Eps] Epstein, D.: Word Processing in Groups. Sudbury, MA: Jones and Bartlett Publishers 1992 [Fal] Falconer, K.: Fractal Geometry, Mathematical Foundations and Applications. Cambridge: Cambridge Univ. Press 1990 [McK] McKean, H.: Selberg’s Trace Formula as Applied to a Compact Riemann Surface. Comm. Pure Appl. Math. 25, 225–246 (1972)
454 [Mil] [Ota] [PP] [Pes] [PeW] [PW] [Rat] [Rue] [Sim] [Sin] [Sun] [Vig] [Wei] [Wol]
M. Pollicott, H. Weiss Milnor, J.: A Note on Curvature and Fundamental Group. J. Diff. Geom. 2, 1–7 (1968) Otal, J.P.: Le Spectre Marqu´e des Longueurs des Surfaces a` Courbure Negative. Ann. of Math. 131, 151–162 (1990) Parry, W., Pollicott, M.: Zeta Functions and the Periodic Orbit Structures of Hyperbolic Dynamics. Ast´erisque 187–188, 1990 Pesin,Y.: Dimension Theory in Dynamical Systems. Chicago Lectures in Mathematics, Chicago, IL: University of Chicago Press, 1997 Pesin, Y., Weiss, H.: The Multifractal Analysis of Gibbs Measures: Motivation, Mathematical Foundation, and Examples. Chaos 7, 89–106 (1997) Pollicott, M., Weiss, H.: Free Energy as a Dynamical Invariant (or Can You Hear the Shape of a potential?). Comm. Math. Physics 240, 457–482 (2003) Ratner, M.: Markov Decomposition for an U-flow on a Three-dimensional Manifold. Mat. Zametki 6, 693–704 (1969) Ruelle, D.: Thermodynamic Formalism. Reading, MA: Addison-Wesley 1978 Simon, B.: The Statistical Mechanics of Lattice Gases. Vol 1, Princeton, NJ: Princeton U. Press, 1993 Sinai, Y.: The Construction of Markov Partitions. Fun. Anal. Appl. 2, 70–80 (1968) Sunada, T.: Riemannian Coverings and Isospectral Manifolds.Ann. of Math. 121, 69–186 (1985) Vigneras, M.F.: Vari´et´es Riemanniennes Isospectrales et non Isometriques. Ann. of Math. 112, 21–32 (1980) Weiss, H.: Spectrum of Equilibrium Measures for Conformal Expanding Maps and Axiom-A Surface Diffeomorphisms. JSP 95, 615–632 (1999) Wolpert, S.: The Length Spectra as Moduli for Compact Riemann Surfaces. Ann. of Math. 109, 323–351 (1979)
Communicated by G. Gallavotti
Commun. Math. Phys. 260, 455–480 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1416-4
Communications in
Mathematical Physics
Dirac Sigma Models Alexei Kotov1,3 , Peter Schaller2 , Thomas Strobl3 1
Max-Planck-Institut f¨ur Mathematik, Bonn, Vivatsgasse 7, 53111 Bonn, Germany. E-mail: [email protected] 2 Operational Risk and Risk Analysis, Bank Austria Creditanstalt, Julius Tandler Platz 3, 1090 Vienna, Austria. E-mail: [email protected] 3 Institut f¨ ur Theoretische Physik, Friedrich-Schiller-Universit¨at Jena, Max-Wienpl. 1, 07743 Jena, Germany. E-mail: [email protected] Received: 16 November 2004 / Accepted: 20 April 2005 Published online: 31 August 2005 – © Springer-Verlag 2005
Abstract: We introduce a new topological sigma model, whose fields are bundle maps from the tangent bundle of a 2-dimensional world-sheet to a Dirac subbundle of an exact Courant algebroid over a target manifold. It generalizes simultaneously the (twisted) Poisson sigma model as well as the G/G-WZW model. The equations of motion are satisfied, iff the corresponding classical field is a Lie algebroid morphism. The Dirac Sigma Model has an inherently topological part as well as a kinetic term which uses a metric on worldsheet and target. The latter contribution serves as a kind of regulator for the theory, while at least classically the gauge invariant content turns out to be independent of any additional structure. In the (twisted) Poisson case one may drop the kinetic term altogether, obtaining the WZ-Poisson sigma model; in general, however, it is compulsory for establishing the morphism property. Contents 1. Introduction . . . . . . . . . . . . . . . 2. From G/G to Dirac Sigma Models . . . 3. Dirac Structures . . . . . . . . . . . . . 4. Change of Splitting . . . . . . . . . . . 5. Field Equations and Gauge Symmetries 6. Hamiltonian Formulation . . . . . . . . References . . . . . . . . . . . . . . . . . .
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1. Introduction In this paper we introduce a new kind of two-dimensional topological sigma model which generalizes simultaneously the Poisson Sigma Model (PSM) [27, 15, 16] and the G/G WZW model [11, 12] and which corresponds to general Dirac structures [8, 22] (in exact Courant algebroids). Dirac structures include Poisson and presymplectic structures as
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particular cases. They are dim M-dimensional subbundles D of E := T ∗ M ⊕ T M satisfying some particular properties recalled in the body of the paper below. If one regards the graph of a contravariant 2-tensor P ∈ (T M ⊗2 ) viewed as a map from T ∗ M to T M, then D = graph P turns out to be a Dirac structure if and only if P is a Poisson bivector (i.e. {f, g} := P(df, dg), f, g ∈ C ∞ (M), defines a Poisson bracket on M). Likewise, D := graph ω, where ω is a covariant 2-tensor viewed as a map T M → T ∗ M is a Dirac structure, iff ω is a closed 2-form. More generally, the above construction is twisted by a closed 3-form H , and in addition not any Dirac structure can be written as a graph from T ∗ M to T M or vice versa. This is already true for a Dirac structure that can be defined canonically on any semisimple Lie group G and which turns out to govern the G/G-WZW model. Only after cutting out some regions in the target G of this σ -model, the Dirac structure D is the graph of a bivector and the G/G model can be cast into the form of a (twisted) PSM [3, 10]. The new topological sigma model we are suggesting, the Dirac Sigma Model (DSM), works for an arbitrary Dirac structure. We remark parenthetically that also generalized complex structures, which lately have received increased attention in string theory, fit into the framework of Dirac structures. In this case one regards the complexification of E and, as an additional condition, the Dirac structure D called a “generalized complex structure” needs to have trivial intersection with its complex conjugate. The focus of this text is on real E, but we intend to present an adaptation separately (for related work cf. also [35, 21, 4]). As is well-known, the quantization of the PSM yields the quantization of Poisson manifolds [18, 6] (cf. also [26]). In particular, the perturbative treatment yields the Kontsevich formula. The reduced phase space of the PSM on a strip carries the structure of a symplectic groupoid integrating the chosen Poisson Lie algebroid [7]. One may expect to obtain similar relations for the more general DSM. Also, several two-dimensional field theories of physical interest were cast into the form of particular PSMs [27, 17, 30, 14] and thus new efficient tools for their analysis were accessible. The more general DSMs should permit to enlarge this class of physics models. The definition of the DSM requires some auxiliary structures. In particular one needs a metric g and h on the target manifold M and on the base or worldsheet manifold , respectively. The action of the DSM consists of two parts, SDSM = Stop + Skin , where only the “kinetic” term Skin depends on g and h. If D = graph P, Skin may be dropped, at least classically, in which case one recovers the PSM (or its relative, twisted by a closed 3-form, the WZ-Poisson Sigma Model). We conjecture that for what concerns the gauge invariant information captured in the model on the classical level one may always drop Skin in SDSM —and for ∼ = S 1 ×R we proved this, cf. Theorem 4 below. Still, even classically, it plays an important role, serving as a kind of regulator for the otherwise less well behaved topological theory. E.g., in general, it is only the presence of Skin which ensures that the field equations of SDSM receive the mathematically appealing interpretation of Lie algebroid morphisms from T to the chosen Dirac structure D—in generalization of an observation for the PSM [5]. (We will recall these notions in the body of the paper, but mention already here that T as well as any Dirac structure canonically carry a Lie algebroid structure). Without Skin , the solutions of the Euler Lagrange equations constrain the fields less in general, which then seems to be balanced by additional gauge symmetries broken by Skin . These additional symmetries can be difficult to handle mathematically, since in part they are supported on lower dimensional regions in the target of the σ -model.
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The paper is organized as follows: In Sect. 2 we use the G/G model as a starting point for deriving the new sigma model. This is done by rewriting the G/G-WZW model in terms suitable for a generalization. By construction, the generalization will be such that the PSM is included, up to Skin , as mentioned above. The role of the Poisson bivector P in the PSM is now taken by an orthogonal operator O on T M, which in the Poisson case is related to P by a Cayley transform, but which works in the general case. In Sect. 3 we provide the mathematical background that is necessary for a correct interpretation of the structures defining the general sigma model. This turns out to be the realm of Courant algebroids and Dirac structures. We recapitulate definitions and facts known in the mathematics literature, but also original results, developed to address the needs of the sigma model, are contained in this section. The action of the Dirac sigma model is then recognized as a particular functional on the space of vector bundle morphisms φ : T → D, SDSM = SDSM [φ]. Specializing this to the PSM, one reproduces the usual fields φ˜ : T → T ∗ M, since precisely in this case D is isomorphic to T ∗ M. In Sect. 4 we point out that the definition of the DSM presented in the preceding sections also depends implicitly on some further auxiliary structure in addition to the chosen Dirac structure D ⊂ E, namely on a “splitting” in the exact Courant algebroid 0 → T ∗ M → E → T M → 0. This dependence occurs in Stop , but again at the end of the day, the “physics” will not be affected by it. In Sect. 5 we derive the field equations of SDSM , which we present in an inherently covariant way. We also prove that φ solves the field equation, iff it respects the canonical Lie algebroid structures of T and the Dirac structure D, i.e. iff φ : T → D is a Lie algebroid morphism. We present one possible covariant (global and frame independent) form of the gauge symmetries of SDSM , furthermore, using the connection on M induced by the auxiliary metric g. We will, however, postpone the corresponding proof of the gauge invariance and further interpretations to another work [20], where the question of covariant gauge symmetries will be addressed in a more general framework of Lie algebroid theories, for which the DSM serves as one possible example. There we will also relate these symmetries to the more standard presentations of the symmetries of the G/G model and the (WZ-)PSM. In Sect. 6, finally, which in most parts can be read also directly after Sect. 2, we determine the Hamiltonian structure of the DSM. In fact, we will do so even for a somewhat more general sigma model, where the target subbundle D ⊂ E is not necessarily required to be integrable. It turns out that the constraints of this model are of the form introduced recently in [2], where now currents J are associated to any section ψ ∈ (D). As a consequence of the general considerations in [2], the constraints Jψ = 0 are found to be first class, iff D is integrable, i.e. iff it is a Dirac structure. 2. From G/G to Dirac Sigma Models We will use the G/G-WZW-model [11, 12] as a guide to the new sigma model that is attached to any Dirac structure. Given a Lie group G with quadratic Lie algebra g and a closed 2-manifold equipped with some metric h, which we assume to be of Lorentzian signature for simplicity, the (multivalued) action functional of the WZW-model consists of two parts (cf. [33] for further details): a kinetic term for G-valued fields g(x) on x as well as a Wess-Zumino term SWZ , requiring the (non-unique) extension of im ⊂ G to a 3-manifold N3 ⊂ G such that ∂N3 = im: k SWZW [g] =
∂+ gg−1 , ∂− gg−1 dx − ∧ dx + + SW Z , (1) 4π
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SWZ [g] =
k 12π
dgg−1 ∧, (dgg−1 )∧2 ,
(2)
N3
x+, x−
are lightcone coordinates on (i.e. h = ρ(x + , x − ) dx + dx − for some where locally defined positive function ρ), ·, · denotes the Ad-invariant scalar product on g, and k is an integer multiple of (which implies that the exponent of i SWZW , the integrand in a path integral, is a unique functional of g : → G). Introducing a connection 1-form a on with values in a Lie subalgebra h < g, one can lift the obvious rigid gauge invariance of (1) w.r.t. g → Adh g ≡ hgh−1 , h ∈ H < G, to a local one (h = h(x) arbitrary, a → hdh−1 + Adh a) by adding to SWZW , k
a+ , ∂− gg−1 − a− , g−1 ∂+ g + a+ , a− Sgauge [g, a] = 2π − a+ , ga− g−1 d2 x , (3) where d2 x ≡ dx − ∧ dx + . For the maximal choice H = G this yields the G/G-model: SG/G [g, a] = SWZW [g] + Sgauge [g, a] .
(4)
In [3] it was shown that on the Gauss decomposable part GGauss of G (for SU (2) this is all of the 3-sphere except for a 2-sphere) the action (4) can be replaced equivalently by a Poisson Sigma Model (PSM) with target GGauss . (This was re-derived in a more covariant form in [10]). It is easy to see that by similar manipulations—and in what follows we will demonstrate this by a slightly different procedure—(4) can be cast into a WZ-PSM [16] on G1 := G\G0 , where G0 = {g ∈ G| ker(1 + Adg ) = {0}} (again a 2-sphere for SU (2)). The question may arise, if there do not exist possibly some other manipulations that can cast the G/G-model into the form of a WZ-PSM globally, with a 3-form H of the same cohomology as the Cartan 3-form (the integrand of (2)). In fact this is not possible: it may be shown [1] that there is a cohomological obstruction for writing the Dirac structure which governs the G/G-model and which is disclosed below (the Cartan-Dirac structure, cf. Example 3 below) globally as a graph of a bivector. Consequently, this calls for a new type of topological sigma model that can be associated to any Dirac structure D (in an exact Courant algebroid) such that it specializes to the WZ-PSM if D may be represented as the graph of a bivector and e.g. to the G/G model if the target M is chosen to be G and D the Cartan-Dirac structure.1 Vice versa, the G/G model already provides a possible realization of the sought-after sigma model for this particular choice of M and D. We will thus use it to derive the new sigma model within this section. For this purpose it turns out to be profitable to rewrite (4) in the language of differential forms, so that the dependence on the worldsheet metric h becomes more transparent; for simplicity we put k = 4π in what follows (corresponding to a particular choice of ). We then find 1 SG/G =
dgg−1 ∧, ∗dgg−1 + SWZ + a ∧, ∗a − a ∧, Adg (∗ − 1)a (5) 2 + a ∧, (∗ − 1)dgg−1 − Adg a ∧, (∗ + 1)dgg−1 . (6) 1
We remark parenthetically that in an Appendix in [3] it was shown that the G/G model can be represented on all of G as what we would call these days a WZ-PSM, but this was at the expense of permitting a distributional 3-form H (the support of which was on G\GGauss ). The above mentioned topological obstruction relies on the smooth category.
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Here our conventions for the Hodge dual operator ∗, which is the only place where h enters, is such that ∗dx ± = ±dx ± . Now we split SG/G into terms containing ∗ and those which do not. One finds that the first type of terms combines into a total square and that SG/G [g, a] = Skin + Stop , 1 Skin = ||dgg−1 + (1 − Adg )a||2 , 2 Stop = −(1 + Adg )a ∧, dgg−1 + a ∧, Adg a + SWZ ,
(7) (8) (9)
where for a Lie algebra valued 1-form β we use the notation ||β||2 ≡ β ∧, ∗β . Before generalizing this form of the action, we show that G/G can be cast into the form of a WZ-PSM (or HPSM for a given choice of H ) on G1 . For this purpose we briefly recall the action functional of the WZ-PSM [16] (cf. also [24]): Given a closed 3-form H and a bivector field P = 21 P ij (X) ∂i ∧ ∂j on a target manifold M one considers SHPSM [X, A] =
Ai ∧ dX + i
1 ij 2 P Ai
∧ Aj +
H,
(10)
N3
where X : → M and A ∈ (T ∗ ⊗ X∗ T ∗ M) and the last term is again a WZ-term (i.e. N3 ⊂ M is chosen such that its boundary agrees with the image of X and the usual remarks about multi-valuedness of the functional can be made)2 ; for the case that H = dB the latter contribution can be replaced by (the single-valued) 21 Bij dX i ∧ dX j . This kind of theory is topological (has a finite dimensional moduli space of classical solutions modulo gauge transformations) iff the couple (P, H ) satisfies a generalization of the Jacobi-identity, namely
P il ∂l P j k + cycl(ij k) = Hi j k P i i P j j P k k ;
(11)
(P, H ) then defines a WZ-Poisson structure on M (also called “twisted Poisson” or “H -Poisson” or “Poisson with background” in the literature). We now want to show that when restricting g to maps g : → G1 , the action SG/G can be replaced by (10) for a particular choice of P and H , at least for what concerns the classical field equations. Let us consider the variation of the two contributions to SG/G in (4) with respect to a separately: δSkin = (1 − Adg ) ∗ dgg−1 + (1 − Adg )a , δa δStop −Adg = (1 + Adg ) dgg−1 + (1 − Adg )a . δa
−Adg
(12) (13)
2 In particular, assume that has no boundary and is orientable. Assume furthermore that H (M) is 2 trivial and that [H ] ∈ H 3 (M, 2π Z). Then one can give meaning to the exponential of iSHPSM [X, A]/, which is the integrand of a “path integral”: Choose any (possibly degenerate) 3-manifold N3 ⊂ M with boundary im(X) to perform the integral. Note that the result iSHPSM [X, A]/ does not only depend on the cohomology class of H , but also the representative. For the field equations of such a “functional” SHPSM , H need not have integral cycles, moreover, and we may drop the conditions [H ] ∈ H 3 (M, 2π Z) and H2 (M) = 0, since only infinitesimal variations are needed. Likewise, a Hamiltonian formulation exists for any closed H, cf. Sect. 6. If has a boundary, additional data need to be specified on “D-branes” in M; for the corresponding Hamiltonian formulation, cf. e.g. [2].
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Decomposing the term in the square bracket into its ±1 eigenvalues of ∗ (use projectors 1 + − 2 (1 ± ∗) or, equivalently, consider the dx and dx components of (13)), it is easy to see that δSG/G /δa = 0 yields dgg−1 + (1 − Adg )a = 0 .
(14)
On the other hand, as obvious from (13), this equation is also obtained from Stop on G1 . Since Skin is quadratic in the left-hand side of (14), it gives no contribution to the variation of SG/G w.r.t. g, which proves the desired equivalence. For completeness we remark that the second field equation is nothing but the zero curvature condition3 F ≡ da + a ∧ a = 0 ,
(15)
and that this equation results from variation of Stop w.r.t. g even on all of G. Note however that Stop will have solutions mapping into G0 = G\G1 violating (14). It thus remains to cast Stop into the form (10). On G1 this is done most easily by introducing A := −(1 + Adg )a, where the matrix components of this 1-form correspond to a right-invariant basis of T ∗ G (note also that sections of T ∗ G and T G, 1-forms and vector fields on the group, can be identified by means of the Killing metric); then the first term in (9) already takes the form of the first term in (10). The WZ-terms can be identified without any manipulations. It remains to calculate the bivector upon comparison of the respective second terms. A very simple calculation then yields P=
1 − Adg , 1 + Adg
H =
1
dgg−1 ∧, (dgg−1 )∧2 , 3
(16)
where P refers to a right-invariant basis on G again and H is the Cartan 3-form. In [29] it was shown that the WZ-Poisson structures [16] are particular Dirac structures; and the utility of this reformulation was stressed due to the simplification of checking (11) for the above example. We want to use the opportunity to stress the usefulness of sigma models in this context (cf. also [31]): Using the well-known fact that the G/G model is topological (in the sense defined above) and that one can cast it into the form (10) is already sufficient to establish (11) for (16); even more, the above consideration is a possible route for finding this example of a WZ-Poisson or Dirac structure. The above bivector also plays a role in the context of D-branes in the WZW-model. We now come to the generalization of the G/G model written in the form (7). For this purpose we first rewrite (10) into a form more suitable to the language of Dirac structures. It is described as the graph of the bivector P in the bundle E = T ∗ M ⊕ T M, i.e. as pairs (α, P(α, ·)) for any α ∈ T ∗ M. With the 1-forms A taking values in T ∗ M we thus may introduce the dependent 1-form V = P(A, ·) taking values in T M. Together they may be viewed as a 1-form A = A ⊕ V on taking values in the subbundle D = graph(P) ⊂ E. Then (10) can be rewritten as i i 1 Stop [X, A] = Ai ∧ dX − 2 Ai ∧ V + H. (17)
N3
Comparison with (9) shows that in the G/G-model Aα = − (1 + Adg )a α , where α is an index referring to a right-invariant basis on G. we can read off Correspondingly, α by comparison of (9) with (17) that then V α = − (1 − Adg )a (showing equality is 3
Written in a matrix representation. More generally, F ≡ da + 21 [a ∧, a].
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a simple exercise where one uses that Adg is an orthogonal operator w.r.t. the Killing metric). Note that in the formulation (17) no metric on M appears anymore; the Killing metric is used only in the above identification. The G/G-model also contains a second part, which uses a metric h on as well as a metric g on M. From the above identification it is easy to generalize it: α Skin [X, A] = ||dX − V ||2 , (18) 2 where for any T M-valued 1-form f = f i ∂i = fµi dx µ ⊗ ∂i on we use ||f ||2 := g(f ∧, ∗f ) ≡ gij hµν fµi fνj vol , vol ≡ det(hµν )d2 x ,
(19)
and where α is some coupling constant. For the action functional of the Dirac Sigma Model (DSM) we thus postulate SDSM [X, A] := Skin + Stop , i.e. α 2 i i 1 SDSM [X, A ⊕ V ] = ||dX − V || + Ai ∧ dX − 2 Ai ∧ V + H . (20) 2 N3 As already mentioned above, the 1-forms A ≡ A ⊕ V take value in X∗ D, where D is a Dirac structure; this will be made more precise and explicit below. For Lorentzian signature metrics h on , α should be real and (preferably) non-vanishing; for Euclidean signatures of h we need an imaginary unit as a relative factor between the kinetic and the topological term. Although possibly unconventional, we will include it in the coupling constant α in front of the kinetic term, so that we are able to cover all possible signatures in one and the same action functional. If g has an indefinite signature, on the other hand, we in addition need to restrict to a neighborhood of the original value α = 1 (or α = i for Euclidean h); the condition we want to be fulfilled is the invertibility of the operator (42) below (cf. also the text following Corollary 2). The metrics h and g on and M, respectively, are of auxiliary nature. First of all, it is easy to see that Skin gives no contribution to the field equations for what concerns the WZ-PSM (10); also the gauge symmetries are modified only slightly by some on-shell vanishing, and thus physically irrelevant contribution (both statements will be proven explicitly in subsequent sections). Let us consider the other extreme case of a Dirac structure provided by the subbundle D = T M to E = T ∗ M ⊕ T M (for H being zero): In this case A ≡ 0 and V = V i ∂i is an unconstrained 1-form field. Obviously in this case Stop ≡ 0 and one obtains no field equations from this action alone. On the other hand, the field equations from Skin are computed easily as dX i = V i .
(21)
First we note that this equation does not depend either on h or on g; these two structures are of auxiliary nature for obtaining a nontrivial field equation in this case, a fact that will be proven also for the general case in the subsequent section (cf. Theorem 1 below). Secondly, we observe that at the end of the day even in this case the two theories Stop [X, A] ≡ 0 and SDSM [X, A] ≡ Skin [X, V ] are still not so different as one may expect at first sight: The moduli space of classical solutions is the same for both theories. The lack of field equations in the first case is compensated precisely in the correct way by additional gauge symmetries, that are absent for Skin [X, V ]. If we permit as gauge symmetries those that are in the connected component of unity, we find the homotopy classes [X] of X : → M as the only physically relevant information.
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(Gauge identification of different homotopy classes might be considered as large gauge transformations, as both action functionals remain unchanged in value; then the moduli space would be just a point in each case). One may speculate that this mechanism of equivalent moduli spaces occurs also in the general situation. We leave this as a conjecture for a general choice of , proving it in the case of = S 1 × R, where we will establish equivalent Hamiltonian structures (cf. Sect. 6 below). We remark, however, that the equivalence may require some slightly generalized notion of gauge symmetries similar to the λ-symmetry discussed in [32]; this comes transparent already from the G/G example, where the additional classical solutions found above, which are located at regions in G where the kernel of 1 + Adg is nontrivial, need to be gauge identified by additional gauge symmetries of Stop that are concentrated at the same region in G. Note that this complication disappears when Skin is added to Stop . Likewise, the field equations (21) have a nice mathematical interpretation; they are equivalent to the statement that the fields (X, A) are in one-to-one correspondence with morphisms from T to D, both regarded as Lie algebroids (cf. Theorem 1 below). So, the addition of Skin (with non-vanishing α) serves as a kind of regulator for the theory, making it mathematically more transparent and more tractable–while simultaneously the “physics” (moduli space of solutions) seems to remain unchanged in both cases, α = 0 and α = 0. Having an auxiliary metric g on M at one’s disposal, it may profitably be used to reformulate Stop . In particular, it will turn out that one may use it to parameterize the Dirac structure globally in terms of an orthogonal operator O : T M → T M (cf. Proposition 1 below), generalizing the operator Adg on M = G in the G/G model above; this then permits one to use unrestricted fields for the action functional again, such as g and a in the G/G model and X and A in the WZ-PSM. Essentially, this works as follows: By means of g we may identify T ∗ M with T M, so ∼ E = T M ⊕ T M and the parts A and V of A = A ⊕ V may be viewed both as (1-form valued) vector (or covector) fields on M (corresponding to the index position, where indices are raised and lowered by means of g). Introducing the involution τ : E → E that exchanges both copies of T M, τ (α ⊕ v) = v ⊕ α, let us consider its eigenvalue subbundles E± = {(v ⊕ ±v) , v ∈ T M}, both of which can be identified with T M by projection to the first factor T ∗ M ∼ = T M. It turns out (cf. Proposition 1 below) that any Dirac structure D ⊂ E can be regarded as the graph of a map from E+ → E− , which, by the above identifications, corresponds to a (point-wise) map O from T M to itself. Let us denote the E+ and E− decomposition of an element of E as (v1 ; v2 ), where elements vi may be regarded as vectors on M. Then any Dirac structure can be written as D = {(v; Ov) ∈ E, v ∈ T M}, where O is point-wise orthogonal w.r.t. the metric g. Obviously, (v; Ov) = (1 + O)v ⊕ (1 − O)v ∈ T M ⊕ T M ∼ = T ∗ M ⊕ T M = E; thus e.g. the graph D = graph(P) of a bivector field P, D = {α ⊕ P(α, ·), α ∈ T ∗ M} ⊂ E corresponds to the orthogonal operator O=
1−P 1+P
⇔
P=
1−O . 1+O
(22)
Note that in a slight abuse of notation we did not distinguish between the bivector field P = 21 P ij ∂i ∧ ∂i ∈ (2 T M), the canonically induced map from T ∗ M → T M, α → P(α, ·), and the corresponding operator on T M using the isomorphism induced by g: T M v → P(g(v, ·), ·) ∈ T M; in particular this implies that in an explicit matrix calculation using some local basis ∂i in T M, with O = Oji ∂i ⊗ dXi , the matrix denoted by P in (22) is Pi j ≡ gik P kj .
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Obviously the Dirac structure of the G/G model corresponds to the choice Adg for O above and the first formula (16) is the specialization of (22) to this particular case. The transformation (22) is a Cayley map. Although any antisymmetric matrix P yields an orthogonal matrix O by this transformation, the reverse is not true. This is the advantage of using O over P, as it works for any Dirac structure.4 Certainly such as the bivector of WZ-Poisson structure has to satisfy an integrability condition, namely Eq. (11), which for H = 0 states that P defines a Poisson structure. There is a likewise condition to be satisfied by Oji (X) so that, more generally, O describes a Dirac structure. O corresponds to a Dirac structure iff U := 1 − O satisfies (cf. Proposition 2 below): ˜
˜
˜
˜
˜
U i i U j j ;i˜ (1 − U )j˜k + cycl(i, j, k) = 21 Hi˜j˜k˜ U i i U j j U k k .
(23)
Here the semicolon denotes the covariant derivative with respect to the Levi Civita connection of g. Locally it may be replaced by an ordinary partial derivative, if the auxiliary metric is chosen to be flat on some coordinate patch. Having characterized D by O ∈ (O(T M)), we may parameterize A ∈ (, X∗ D) more explicitly by a = a i ∂i ∈ (, X ∗ T M) according to A = −(1+O)a⊕−(1−O)a. Then the total action (20) can be rewritten in the form α 2 ∧ ∧ ||dX + (1 − O)a|| + g(dX , (1 + O)a) + g(a , Oa) + H SDSM [X, a] = 2 N3 α j m i i i k j j ≡ (dX + a − Ok a ) ∧ ∗(dX + a − Om a ) gij (24) 2 + dX i ∧ a j (gij + Oij ) + Oij a i ∧ a j + H,
N3
where now X i and a i , local 0-forms and 1-forms on , respectively, can be varied without any constraints and indices are lowered and raised by means of gij (X) and gij (X), respectively. We stress again that the g-dependence of the last line is ostensible only, whereas the α-dependent terms depend on it inherently. The above presentation of A was suggested by the G/G model. In the rest of the paper we will however rather use the slightly more elegant parameterization A = (1 + O)a ⊕ (1 − O)a, resulting from a := −a. In these variables the action (24) takes the form α SDSM [X, a] = ||f ||2 + g((1 + O)a ∧, dX) + g(a ∧, Oa) + H, (25) 2
N3
where we used (19) with f ≡ dX − V ≡ dX − (1 − O)a .
(26)
In the following sections we will tie the above formulas to a more mathematical framework and, among others, analyze the field equations from this perspective. Readers more interested in applications for physics may also be content with consulting only 4 This observation is due to the collaboration of A.K. and T.S. with A. Alekseev and elaborated further in [1]. We are also grateful to A. Weinstein for pointing out to us that the description of Dirac structures by means of sections of O(T M) was used already in the original work [8].
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the main results from the following sections, in particular Theorem 1 and Proposition 7, and then turn directly to the Hamiltonian analysis of the action in Sect. 6. Maybe also noteworthy is the generalization Eq. (44) of the kinetic term introduced in Sect. 4 below. The modification of the old kinetic term uses a 2-form C on the target and is independent of any metric. Such a generalized kinetic term is suggested by the more mathematical considerations to follow, but will not be pursued any further within the present paper. We close this section with a continuative remark: As mentioned above, for non-vanishing parameter α, the classical theory will turn out not to depend on this parameter. It is tempting to believe that this property can be verified also on the quantum level, a change in α corresponding to the addition of a BRS-exact term. In this context it may be interesting to regard the limit α → ∞, yielding localization to f = 0. In fact, one may expect localization of the path integral to all equations of motion, cf. [34, 13]. 3. Dirac Structures The purpose of this section is to provide readers with the mathematical background for the structures used to define the Dirac sigma model. We review some basic facts about Dirac structures, being maximally isotropic (Lagrangian) subbundles in an exact Courant algebroid, the restriction of the Courant bracket to which is closed. We describe an explicit isomorphism between the variety of all Lagrangian subbundles and the group of point-wise acting operators in the tangent bundles, orthogonal with respect to a fixed Riemann metric. We derive an obstruction for such operators to represent a Dirac structure, cf. Proposition 2 below. A Courant algebroid [22, 8] is a vector bundle E equipped with a non-degenerate symmetric bilinear form , , a bilinear operation ◦ on (E) (sometimes also denoted as a bracket [·, ·]), and a bundle map ρ : E → T M satisfying the following properties: 1. 2. 3. 4. 5.
The left Jacobi condition e1 ◦ (e2 ◦ e3 ) = (e1 ◦ e2 ) ◦ e3 + e2 ◦ (e1 ◦ e3 ), Representation ρ(e1 ◦ e2 ) = [ρ(e1 ), ρ(e2 )], Leibniz rule e1 ◦ f e2 = f e1 ◦ e2 + Lρ(e1 ) (f )e2 , e ◦ e = 21 D e, e , Ad-invariance ρ(e1 ) e2 , e3 = e1 ◦ e2 , e3 + e2 , e1 ◦ e3 , ρ∗
d
where D is defined as D : C ∞ (M) → 1 (M) → E ∗ E. Properties 2 and 3 can be shown to follow from the other three properties, which thus may serve as axioms (cf. e.g. [19]). A Courant algebroid is called exact [28], if the following sequence is exact: ρ∗
ρ
0 → T ∗ M → E → T M → 0.
(27)
A Dirac structure D in an exact Courant algebroid is a maximally isotropic (or Lagrangian) subbundle with respect to the scalar product, which is closed under the product. A Dirac structure is always a particular Lie algebroid: By definition a Lie algebroid is a vector bundle F → M together with a bundle map ρ : F → T M and an antisymmetric product (bracket) between its sections satisfying the first three properties in the list above (where again the second property can be derived from the other two). In particular, the product, often also denoted as a bracket, e1 ◦ e2 :≡ [e1 , e2 ], defines an infinite dimensional Lie algebra structure on (F ). Due to the isotropy of D, the induced product (bracket) becomes skew-symmetric and obviously D is a Lie algebroid.
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From now on we will only consider exact Courant algebroids (27). Let us choose a “connection” on E, i.e. an isotropic splitting σ : T M → E, ρ ◦ σ = id. The difference σ (X) ◦ σ (Y ) − σ ([X, Y ]) = ρ ∗ H (X, Y )
(28)
is a pull-back of a C ∞ (M)−linear, completely skew-symmetric tensor H ∈ 3 (M), given by H (X, Y, Z) = σ (X) ◦ σ (Y ), σ (Z) . From the above axioms one may deduce the “Bianchi identity”: dH = 0. Once a connection is chosen, any other one differs by the graph of a 2−form B. Its curvature is equal to H + dB. Therefore the cohomology class [H ] ∈ H 3 (M) is completely determined by the Courant algebroid [28]. Choosing a splitting with the curvature 3-form H , it is possible to identify the corresponding exact Courant algebroid with T ∗ M ⊕ T M and the scalar product with the natural one:
ξ1 + θ1 , ξ2 + θ2 = θ1 (ξ2 ) + θ2 (ξ1 ) ,
(29)
where ξi ∈ (T M), θi ∈ 1 (M). Finally, the multiplication law can be shown to take the form: (ξ1 + θ1 ) ◦ (ξ2 + θ2 ) = [ξ1 , ξ2 ] + Lξ1 θ2 − ıξ2 dθ1 + H (ξ1 , ξ2 , ·) ,
(30)
where Lξ and iξ denote the Lie derivative along a vector field ξ and contraction with ξ , respectively. Let E be an exact Courant algebroid with a chosen splitting E = T ∗ M ⊕ T M and a vanishing 3-form curvature (this implies that the chacteristic 3-class of E is trivial). Then we have: Example 1. Let D be a graph of a Poisson bivector field P ∈ (2 T M) considered as a skew-symmetric map from T ∗ M to T M, then D = {θ ⊕ P(θ )} is a Dirac subbundle and the projection from D to T ∗ M is bijective. Any Dirac subbundle of E with bijective projection to T ∗ M is a graph of a Poisson bivector field. Example 2. Let D be a graph of a closed 2−form ω ∈ 2 (M) considered as a skewsymmetric map from T M to T ∗ M, then D = {ω(v) ⊕ v} is a Dirac subbundle and the projection of D to T M is bijective. Any Dirac subbundle of E for which the projection to T M is bijective is a graph of a closed 2−form. If E has a non-trivial characteristic class [H ], one needs to replace the Poisson (presymplectic) structure by a corresponding WZ (or twisted) one, cf. [16, 29]. Now we describe all Lagrangian subbundles of an exact Courant algebroid E, which are not necessarily projectable, either to T ∗ M (for any splitting σ ) or to T M.5 Let us choose an arbitrary Riemannian metric g on M, which can be thought of as a non-degenerate symmetric map from T M to T ∗ M. The inverse of g is acting from T ∗ M to T M. We denote these actions as ξ → ξ ∗ and θ → θ ∗ for a vector field ξ and a 1-form θ, respectively. In some local coordinate chart it can be written as follows: ∗ ∗ (31) ξ ∗ = ξ i ∂i = ξ i gij dx j , θ ∗ = θi dx i = θi gij ∂j . 5 At least most of the rest of the section seems to be original, taking into account, however, the previous footnote for what concerns Proposition 1 below.
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Given a splitting σ , one can combine these maps to a bundle involution τ : E → E, θ ⊕ ξ → ξ ∗ ⊕ θ ∗ , with the obvious property τ 2 ≡ 1. To simplify the notation, we will henceforth just write θ +ξ instead of θ ⊕ξ , because the nature of θ and ξ anyway indicate the position in E ∼ = T ∗ M ⊕ T M. The bundle E thus decomposes into ±1−eigenvalue parts, E = E+ ⊕ E− , where E± := Ker(τ ∓ 1). Proposition 1. Any Lagrangian subbundle is a graph of an orthogonal map E+ → E− , which can be identified with a section O ∈ (O(T M)). Proof. First, let us show that τ is symmetric with respect to , . In fact, by the definition of the scalar product (29) and τ we have
τ (ξ1 + θ1 ), ξ2 + θ2 = θ2 , θ1∗ + ξ1∗ , ξ2 = g(ξ1 , ξ2 ) + g(θ1 , θ2 ) . Now it is easy to see that the restriction of , to E+ (E− ) is a positive (negative) metric, respectively, and E+ , E− ≡ 0. Therefore we conclude that any Lagrangian subbundle has a trivial intersection with E± . Hence the projection of D to E+ is bijective which implies that D is a graph of some map E+ → E− . Let us identify E± with T M by means of ±ρ, then the map uniquely corresponds to an orthogonal transformation O of T M. More precisely, any section u± of E± can be uniquely represented as ξ ± ξ ∗ for some vector field ξ . Now the definition of O yields that any section of D has the form (1 − O)ξ + ((1 + O)ξ )∗ for a certain vector field ξ . Taking into account that , vanishes on D, we show that O is an orthogonal map:
(1 − O)ξ + ((1 + O)ξ )∗ , (1 − O)ξ + ((1 + O)ξ )∗ = g((1−O)ξ, (1+O)ξ ) + g((1+O)ξ, (1−O)ξ ) = 2(ξ, ξ ) − 2g(Oξ, Oξ ) = 0. For the argumentation above, in particular for the fact that D has a trivial intersection with E± , it was important that g is a metric of definite signature. Note, however, that even when g is an arbitrary pseudo-Riemannian metric, we obtain a maximally isotropic subbundle D from a graph in E+ , E− of an pseudo-orthogonal operator O; just not all such subbundles D can be characterized in this way. This is an important fact when we want to cover, e.g., the G/G-model for non-compact semi-simple Lie groups. Locally not any Dirac structure D admits a splitting σ such that D corresponds to either a Poisson or a presymplectic structure. But even if it does so locally, there may be global obstructions for it to be WZ-Poisson or WZ-presymplectic. This can be shown by constructing characteristic classes associated to a given Dirac structure D ⊂ E [1]. An example for such a Dirac structure with “non-trivial winding” is the following one:6 Example 3. Take M = G a Lie group whose Lie algebra g = Lie G is quadratic, with the non-degenerate ad-invariant scalar product denoted by ·, · . Then the respective exact Courant algebroid E = T ∗ G ⊕ T G can be cast into the following form: E = G × (g ⊕ g), ρ(x, y) = x R ≡ xg, .
(x, y), (x , y ) =
x, y + x , y , (x, y) ◦ (x , y ) = (−[x, x ], [x, x ] − [x, y ] + [x , y]), ∀const. sections x, x , y, y 6 This example can be extracted directly from the previous section, cf. the text after formula (16)—we only changed from a right-invariant basis to a left-invariant one.
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The cotangent bundle T ∗ G is embedded as follows: θ → (0, θ ∗ g−1 ), where θ ∗ is the vector field dual to the 1-form θ via the Killing metric on G which is left- and right-invariant. Note that for any left or right invariant vector field ξ one has Lξ (θ )∗ = (Lξ θ)∗ . It is easy to see that the curvature H of the splitting (connection) σ : ξ → (ξg −1 , 0) equals the Cartan 3-form H (ξ1 , ξ2 , ξ3 ) = ξ1 g−1 , [ξ2 g−1 , ξ3 g−1 ] , for ξi ∈ (T G). The natural Dirac structure, considered in Sect. 2, is determined by O = Adg . One can calculate the product of two section of this Dirac structure in the representation defined above (here x, y are constant sections of G × g): ((1 − O)x, (1 + O)x) ◦ ((1 − O)y, (1 + O)y) = (−(1 − O)[x, y], −(1 + O)[x, y]) . (32) Certainly, closure on (D) of the induced product or bracket requires some additional property of the operator O, generalizing e.g. the Jacobi identity of the Poisson bivector in Example 1. Proposition 2. A Lagrangian subbundle, represented by an orthogonal operator O as the set D = {(1 − O)ξ + ((1 + O)ξ )∗ }, is a Dirac structure, iff the following property holds, where ∇ denotes the Levi-Civita connection on M and ξi ∈ (M, T M): 1
g O−1 ∇(1−O)ξσ (1) (O) ξσ (2) , ξσ (3) = H ((1−O)ξ1 , (1−O)ξ2 , (1−O)ξ3 ) . 2 σ ∈Z3
(33) Proof. First, we rewrite the multiplication law in terms of the Levi-Civita connection: x1 ◦ x2 = ∇ρ(x1 ) x2 − ∇ρ(x2 ) x1 + ∇x1 , x2 + H (ρ(x1 ), ρ(x2 ), ·) ,
(34)
where x, y ∈ E, ∇x is thought of as a 1-form taking values in E, and hence ∇x1 , x2 is in 1 (M) ⊂ E. Let us take xi ∈ (M, D), i = 1, 2, 3, written in the form xi = (1 − O)ξi + ((1 + O)ξi )∗ .
(35)
Note that ρ(xi ) = (1 − O)ξi . Using (34), we derive the product x1 ◦ x2 and the 3-product
x1 ◦ x2 , x3 , which is a C ∞ (M)−linear form vanishing if and only if D is closed with respect to the Courant multiplication, ∗ x1 ◦ x2 = (1−O) ∇ρ(x1 ) ξ2 − ∇ρ(x2 ) ξ1 + (1+O) ∇ρ(x1 ) ξ2 − ∇ρ(x2 ) ξ1 + H (ρ(x1 ), ρ(x2 ), ·) − ∇ρ(x1 ) (O)ξ2 + ∇ρ(x2 ) (O)ξ1 ∗ (36) + ∇ρ(x1 ) (O)ξ2 − ∇ρ(x2 ) (O)ξ1 − 2g O−1 ∇(O)ξ1 , ξ2 . In the above we used that the Levi-Civita connection commutes with τ , i.e. ∇ξ (η∗ ) = ∗ ∇ξ η .Apparently, the sum of the first and second terms in (36) belongs to the same maximally isotropic subbundle, therefore its product with x3 vanishes, and x1 ◦ x2 , x3 = (I ) + (I I ) + (I I I ), where ∗ (I ) = −∇ρ(x1 ) (O)ξ2 + ∇ρ(x1 ) (O)ξ2 , (1−O)ξ3 + ((1+O)ξ3 )∗ − (1 ↔ 2) = g −∇ρ(x1 ) (O)ξ2 , (1+O)ξ3 + g ∇ρ(x1 ) (O)ξ2 , (1−O)ξ3 − (1 ↔ 2) = −2g O−1 ∇ρ(x1 ) (O)ξ2 + O−1 ∇ρ(x2 ) (O)ξ3 , ξ1 ,
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and
(I I ) = −2g O−1 ∇(O)ξ1 , ξ2 , x3 = −2g O−1 ∇ρ(x3 ) (O)ξ1 , ξ2 , (I I I ) = H (1 − O)ξσ (1) , (1 − O)ξσ (2) , (1 − O)ξσ (3) .
In the formulas above (1 ↔ 2) denotes the permutation of the first two indices and Z3 is the group of cyclic permutations of order 3. We also used that the orthogonality of O implies that O−1 ∇(O) is a skew-symmetric operator with respect to the metric g, i.e. g(O−1 ∇(O)η1 , η2 ) = −g(O−1 ∇(O)η2 , η1 ) holds for any couple of vector fields η1 , η2 . All in all we then obtain
x1 ◦ x2 , x3 = −2 g O−1 ∇(1−O)ξσ (1) (O), ξσ (2) , ξσ (3) σ ∈Z3
+H ((1 − O)ξ1 , (1 − O)ξ2 , (1 − O)ξ3 ) , which implies (33).
(37)
From the above proof we extract the following useful Corollary 1. Assume that the integrability condition (33) holds and that xi ∈ (D), parameterized as in (35). Then their Courant product (36) can be written as x1 ◦ x2 = (1 − O)Q(ξ1 , ξ2 ) + ((1 + O)Q(ξ1 , ξ2 ))∗ ,
(38)
where ∗ 1 Q(ξ1 , ξ2 ) = ∇ρ(x1 ) ξ2 −∇ρ(x2 ) ξ1 + g ξ1 , O−1 ∇(O)ξ2 + H (ρ(x1 ), ρ(x2 ), ·)∗ , 2 (39) and ρ(xi ) ≡ (1−O)ξi . At the expense of introducing an auxiliary metric g on M, a Dirac structure can be described globally by O ∈ (O(T M)). The introduction of O permits also to identify D with T M (via Eq. (35)). The Courant bracket thus induces a Lie algebroid bracket on D. This in turn induces an unorthodox Lie algebroid structure on T M, where the bracket between two vector fields ξ , ξ is given by [ξ, ξ ] := Q(ξ, ξ ), which defines a Lie algebroid structure on F := T M with anchor ρ : F → T M, ξ → (1 − O)ξ . In a holonomic frame, the corresponding structure functions, [∂i , ∂j ]F = Cijk ∂k , are easily computed as 1 k n Cijk = (1−O)m i mj − (i ↔ j ) + Om j ;k Omi + Hmn k (1−O)m i (1−O)j . 2
(40)
For practical purposes it may be useful to know how the orthogonal operator O transforms when changing g: Proposition 3. Given a fixed splitting so that E = T ∗ M ⊕ T M, the couples (g, O)
describe the same Dirac structure D on M iff and (g, O) −1
= O − 1 + O g−1 g(1 + O) 1 − O + g−1 g(1 + O) .
(41)
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Proof. An arbitrary element ω ⊕ v in D can be parameterized as g(1 + O)ξ ⊕ (1 − O)ξ
for some Equating this g(1 + O) ξ ⊕ (1 − O) ξ , it is elementary to derive ξ ∈ T M.−1 to 1
ξ = 2 1 − O + g g(1 + O) ξ . Since both of these two parameterizations are oneto-one (see Proposition 1), the dependence above is invertible. Using this relation in
equating (1 − O)ξ to (1 − O) ξ for all ξ , we prove the statement of the proposition. As a simple corollary one obtains the following Lemma 1. For any orthogonal operator O and positive or negative symmetric operator b, the operator 1 − O + b(1 + O) is invertible. Both assumptions in the lemma refer to a definite metric (as this was assumed and necessary for an exhaustive description of Dirac structures—cf. the discussion following Proposition 1). For later use we conclude from this Corollary 2. The operator Tα := 1 + O + α(1 − O)∗
(42)
on T ∗ ⊗ X ∗ T M is invertible. Here ∗ is the Hodge operator on T ∗ ; for Lorentzian signatures of h, ∗2 = id and, by assumption, α ∈ R\0, for Euclidean signatures, ∗2 = −id and iα ∈ R\0. The statement above follows in an obvious way from Lemma 1, i.e. for definite metrics g. For pseudo-Riemannian metrics g, however, it in general becomes necessary to restrict α to a neighborhood of α = 1 and α = i for Lorentzian and Euclidean signature of h, respectively. 4. Change of Splitting The action SDSM of the Dirac Sigma Model consists of two parts, the topological term (17) and the kinetic one (18). It was mentioned repeatedly that only the second contribution depends on the auxiliary metrics g and h. However, also the first part Stop (and in fact now only this part) depends on another auxiliary structure, namely the choice of the splitting. We will show in the present section that this dependence is rather mild: It can be compensated by a coordinate transformation on the field space, which is trivial on the classical solutions (cf. Proposition 5 below; the transformation is α-dependent, so it changes if also the kinetic term is taken into account). There is also an interesting alternative: Recall that h−1 ⊗ X∗ g was used as a symmetric pairing in (T ∗ ⊗ X∗ T M) to define Skin . If in addition we are given a 2-form C on M, we can also use h−1 (vol ) ⊗ X ∗ C for a symmetric pairing, where vol is the volume 2-form on induced by h, and h−1 (vol ) denotes the corresponding bivector resulting from raising indices by means of h. Using the sum of both (or, more precisely, an α-dependent linear combination of them) to define Skin , cf. Eq. (44) below, a change of splitting can be compensated by a simple change of the new background field C. From Sect. 3 one knows that a splitting in an exact Courant algebroid is governed by 2-forms B. Namely, assume that σ : T M → E is a splitting, then any other one sends a vector field ξ to σB (ξ ) := σ (ξ ) + B(ξ, ·). Proposition 4. The DSM action transforms under a change of splitting σ → σB according to: 1 SDSM := SDSM + Bij f i ∧ f j , (43) SDSM → 2
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where f i ≡ dXi − V i . Proof. In fact, the decomposition A = A + V is not unique and depends on the splitting.
+ V , Changing the splitting by a 2-form B, we get a different decomposition: A = A
where A = A − B(V , ·). To argue for this we note that V = σ (ρ(A)) is indeed invariant (in particular, Skin is invariant), only A varies, hence after the change of splitting
= A − σB (ρ(A)) = A − B(V , ·). Taking into account that the B-field we obtain A
= H + dB, we calculate: influences H , H → H 1 i i
Stop = Ai ∧ dX − Ai ∧ V + H 2 N3 1 = Stop + B(dX ∧, dX) − 2B(V ∧, dX) + B(V ∧, V ) , 2
which finally gives the required derivation (43).
As mentioned above, if the kinetic term Skin (8) is replaced by 1 new Skin := α g(f ∧, ∗f ) + C(f ∧, f ) 2
(44)
for some auxiliary C ∈ 2 (M), then a change of splitting governed by the B-field merely leads to C → C − B for this new background field. Note that despite the fact that H and C change in a similar way w.r.t. a change of splitting, H → H + dB, C → C − B, they enter the sigma model qualitatively in quite a different way: H is already uniquely given by the Courant algebroid and a chosen splitting, while C is on the same footing as g or h, which have to be chosen in addition. In what follows we will show that a change of splitting does not necessarily lead to a change in the background fields, but instead can also be compensated by a transformation of the field variables—at least infinitesimally. Proposition 5. Let α = 0 and B be a “sufficiently small” 2-form. Then there exists a change of variables a¯ := a + δa such that SDSM [X, a¯ ] ≡ SDSM [X, a] + 21 B(f ∧, f ). Clearly, δa needs to vanish for f = 0. We remark that this equation is one of the field equations, cf. Theorem 1 below, so that δa corresponds to an on-shell-trivial coordinate transformation (on field space). Proof. We find that after the change of variables a¯ := a + δa one has 1 SDSM [X, a¯ ] − SDSM [X, a] = g(Tα δa ∧, f ) + g(δa ∧, Rα δa) , 2
(45)
where Tα is given by Eq. (42) and Rα = O − O−1 + α(2 − O − O−1 )∗ . (46) 1 Solving the equation SDSM [X, a¯ ] − SDSM [X, a] = 2 B(f ∧, f ), we use the ansatz δa = Tα−1 Lf for some yet undetermined L ∈ (, End(T ∗ ⊗ X ∗ T M)) (invertibility of Tα follows from Corollary 2). This yields the following equation for L: L∗ AL + L∗ − L + B = 0 ,
(47)
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where A = Tα−1∗ Rα∗ Tα−1 and B denotes the operator via the identification B(a ∧, b) = g(Ba ∧, b), i.e. the operator is obtained from the bilinear form by raising the second index. In the above the adjoint L∗ of an operator L in T ∗ ⊗X ∗ T M is defined by means of the canonical pairing induced by g: g(a ∧, Lb) = g(L∗ a ∧, b).7 For sufficiently small B, the above operator (or matrix) equation (47) has the following solution: L=
∞
n=0
1 2
n+1
(BA)
n
(48)
B,
where λn ≡ λ(λ−1)...(λ−n+1) . Note that the L above is anti-selfadjoint (antisymmetric), n! L∗ = −L, so it solves the simplified equation LAL + 2L − B = 0. For the case that A has an inverse, the above √ solution can also be rewritten in the more transparent form L = A−1 1 + AB − 1 . Since A can be seen to be bounded, the sum (48) converges for small enough B. For completeness we also display how O transforms under a change of splitting: Proposition 6. Given two splittings σ, σ of the exact Courant algebroid E, the couples
describe the same Dirac structure D on M, iff (σ, O) and ( σ , O)
= (2O − B(1 − O)) (2 − B(1 − O))−1 , O
(49)
where B is defined as follows: σ (ξ ) = σ (ξ ) + (Bξ )∗ for any vector field ξ . Proof. Straightforward calculations similar to Proposition 3.
5. Field Equations and Gauge Symmetries In this section we compute the equations of motion of the Dirac sigma model (DSM) introduced in Sect. 2 and reinterpret them mathematically. In particular we will show that the collection (X, A) of the fields of the DSM are solutions to the field equations if and only if they correspond to a morphism from T to the Dirac structure D, viewed as Lie algebroids. As a consequence they are also independent of the choice of metrics used to define the kinetic term Skin of the model as well as of the splitting used to define Stop . Definition 1 ([23, 5]). A vector bundle morphism φ : E1 → E2 between two Lie algebroids with the anchor maps ρi : Ei → T Mi is a morphism of Lie algebroids, iff the induced map : (E2∗ ) → (E1∗ ) is a chain map with respect to the canonical differentials di : d1 − d2 = 0 .
(50)
7 For operators commuting with the Hodge dual operation ∗ (which applies to all operators appearing here), this coincides with the adjoint defined by the symmetric pairing induced by g and h: g(a ∧, ∗Lb) = g(L∗ a ∧, ∗b). One may then verify e.g. Rα∗ = −Rα , ∗∗ = −∗, A∗ = −A, and Tα∗ = O−1 Tα . This notation is not to be confused with the isomorphism between T M and T ∗ M, extended to an involution τ : T M ⊕ T ∗ M, denoted by the same symbol, cf. Eq. (31).
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First, notice that, fixing a base map X : M1 → M2 , any vector bundle morphism is uniquely determined by a section a ∈ (M1 , E1∗ ⊗ X ∗ E2 ). Hence the morphism property (50) should have a reformulation in terms of the couple (X, a). Second, it is easy to see that the property (50) is purely local, therefore it admits a description for any local frame. Indeed, let {ei } be a local frame of the vector bundle E2 , {ei } be its dual, and a i := (ei ), then (50) is equivalent to the following system of equations: d1 X − ρ2 (a) = 0, 1 d1 a k + Cijk a i ∧ a j = 0. 2
(51) (52)
The first equation is covariant; it implies that the following commutative diagram holds true: φ
E1 −→ E2 ρ1 ↓ ρ2 ↓ X∗
T M1 −→ T M2 The second equation depends on the choice of frame. However, the additional contribution, which arises in (52) under a change of frame, is proportional to the d1 X − ρ2 (a), hence it is of no effect, if Eq. (51) holds. For further details about this definition we refer to [5]. In our context E1 = T and thus d1 becomes the ordinary de Rham differential. The a above becomes A ∈ 1 (, X ∗ D), but can be identified with a due to A ≡ A + V = ((1 + O)a)∗ + (1 − O)a ,
(53)
where a ∈ 1 (, X ∗ T M) is an unrestricted field. In these variables SDSM has the form (25). The first morphism property, Eq. (51), then takes the form f ≡ dX − V = 0. Theorem 1. Let α = 0 (depending on the signatures of h and g, possibly further restricted as specified after Eq. (24) or at the end of Sect. 3). Then the field equations of SDSM have the form f ≡ dX − (1 − O)a = 0 , ∗ 1 1 ∧ −1 (a) + g a , O ∇(O)a + H ((1 − O)a ∧, (1 − O)a, ·)∗ = 0 , 2 4
(54) (55)
or, in the dependent (A, V ) variables, dX = V and 1 1 1 A + g V ∧, O−1 ∇(O)V + A ∧, O−1 ∇(O)A∗ + H (V ∧, V ∧, ·) = 0 . (56) 4 4 2 The fields (X, A) are a solution of the equations of motion, if and only if they induce a Lie algebroid morphism from T to D. Corollary. The classical solutions of the DSM do not depend on the choice of the coupling constant α = 0 (in the permitted domain), or, more generally, on the choice of metrics g and h.
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Proof. Using that O is an orthogonal operator w.r.t. g, one computes in a straightforward generalization of (12) and (13)8 O
δSDSM δa
∗
= (1 + O + α(1 − O)∗) f .
(57)
The term in the brackets is the operator (42), which, according to our assumption on α, is invertible; so, indeed f = 0. Note, however, that in general 1 + O is invertible only if the Dirac structure corresponds to a graph of a bivector. Thus, only in the WZ-Poisson case one may drop the kinetic term altogether if one wants to keep the morphism property of the field equations. We now turn to the X-variation of SDSM . This is conceptually more subtle since a ∈ 1 (, X∗ T M) depends implicitly on X as well. Thus to determine δX a we need a connection, since, heuristically, we are comparing sections in two different, but nearby bundles X0∗ T M and (X0 + δX)∗ T M. (If we required e.g. δX ai = 0, then this would single out a particular holonomic frame, since a change of coordinates on M yields
ai = Mji (X)aj . In the following we develop an inherently covariant formalism that also produces covariant field equations.) Let us denote the local basis in X ∗ T M dual to dX i in X ∗ T ∗ M by ∂ i . The notation dX i is used so as to distinguish it from d acting on the pull-back function X ∗ X i , which we denote as usual by dXi . Then j
δX ∂ i = ki δX k ∂ j ,
(58)
where ji k are coefficients of the Levi Civita connection ∇ of g. Also, we think of ai to depend on both X(x) and x (cf. also [5] for further details); correspondingly, i i j k δX a = δX (a ) + kj a δX ∂ i . Note that certainly δX d = dδX , where d denotes the de Rham operator. However, this does not apply for dX used above, which is the section in T ∗ ⊗ X∗ T M corresponding to the bundle map X∗ : T → T M, dX = dX i ⊗ ∂i ; so in dX, d does not denote an operator. Here one finds in analogy to δX a, i δX (dX) = d(δX i ) + kj dX j δX k ∂ i = (δX) ,
(59)
where is the pull-back of the Levi-Civita connection ∇ to X∗ (T M) and in the last i = i , was used. The above covariant form of the step torsion freeness of ∇, kj jk variation (58) implies in particular that δX g = 0, when g is viewed as an element in (, X∗ T ∗ M ⊗2 )—as it appears in the action functional (24), so we will be permitted to use δX g(a ∧, b) = g(δX a ∧, b) + g(a ∧, δX b) below. With the above machinery at hand, the variation w.r.t. X is rather straightforward again. By construction it will produce a covariant form of the respective field equations. Since we already know that f = 0 holds true, moreover, we will be permitted to drop all terms below which are proportional to f . Correspondingly, Skin can be dropped for the Recall that a = −a and O = Adg in the G/G model. The variational derivative is defined according ∗ ∧ δSDSM . Alternatively, one may infer this relation to δa SDSM = δ a ∧, δSδDSM a ≡ g δa , δa 8
also from Eq. (45), keeping only the terms quadratic in δ a: since g(Tα δ a ∧, f ) = g(δ a ∧, O−1 Tα f ), cf. footnote.
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calculation of δX SDSM = 0, since Skin is quadratic in f . By convention, we will denote equalities up to f = 0 in what follows by ≈; so, in particular δStop δSDSM ≈ . (60) δX δX Also, we may drop all terms containing δX a, since on behalf of (57), they will be proportional to f again, with or without the kinetic term included (corresponding only to α = 0 and α = 0, respectively, in formula (57)). Thus, δX Stop ≈ g (δX O)a ∧, dX + g (1 + O)a ∧, δX
1 +g a ∧, (δX O)a + Hij k dX i ∧ dX j δX k , (61) 2 where we already made use of Eq. (59). In the second term we perform a partial integration (dropping eventual boundary terms) and observe that ((1 + O)a) ≈ 2 a ,
(62)
⊗ ∂ k ≡ 0. which follows from 2a ≈ (1 + O)a + dX and (dX) = Replacing dX by a − Oa in the first term, we then obtain 1 δX Stop ≈ g a ∧, (O−1 δX O)a + g 2 a ∧, δX + Hij k dX i ∧ dX j δX k , (63) 2 −dXi
∧ dX j
jki
where the first and the third term in (61) combined into the first term above. This proves Eq. (55). Equivalence with Eq. (56) is established easily as follows: For the first term we read Eq. (62) from right to left and use A∗ = (1 + O)a. For the second term of (55) we replace a by 21 (A∗ + V ) and utilize the antisymmetry of O−1 ∇O to cancel off-diagonal terms. For the third one we use V = (1 − O)a. This leaves us with proving the equivalence of (55) to the second morphism property (52), specialized to the present setting, where, again, f = 0 may be used freely. So, we need to show that (55) can be replaced by dai + 21 Cji k aj ∧ ak ≈ 0, where the structure functions are given by Eq. (40). Since Cji k aj ∧ ak ⊗ ∂ i ≡ Q(∂j , ∂ k ) aj ∧ ak , most of the terms in (55) are identified easily, and it only remains to show that i ⊗ ∂ i , (64) a ≡ dai + ji k dX j ∧ ak ⊗ ∂ i ≈ dai + aj ∧ak (1−O)m j mk which is an obvious identity.
Having established that the field equations enforce a Lie algebroid morphism T → D, it is natural to expect that on solutions the gauge symmetries correspond to a homotopy of such morphisms [5]. This is indeed the case. For the gauge invariance of an action functional, however, an off-shell (and preferably global) definition of the symmetries is needed. This is a more subtle question. A derivation of the gauge symmetries, including a specialization to known examples, requires some more extended discussion for a sufficiently clear presentation. Moreover, the discussion fits into a more general framework, following the considerations in [5], and thus will be presented elsewhere [20]. In the present paper we only provide the result of such an analysis:9 9 Note, however, that e.g. the fact that the model contains no propagating degrees of freedom (is “topological”) follows also from the self-contained Hamiltonian analysis in the subsequent section.
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Proposition 7. For nonvanishing α the infinitesimal gauge symmetries of SDSM can be expressed in the following form δ X = (1 − O) , ∗ 1 δ a = () − g O−1 ∇(O)a, + H ((1 − O)a, (1 − O), ·)∗ 2 1 + Tα−1 H (f, (1 − O), ·)∗ + (1 − α∗)∇f (O) + Mf , 2
(65)
(66)
where ∈ (, X∗ T M) and M = M ∗ ∈ (End(T ∗ ⊗ X ∗ T M)) may be chosen freely. The operator M above parametrizes trivial gauge symmetries. In the G/G and in the Poisson case the above symmetries reproduce the known ones for α = 1 and α = 0, respectively. In general, however, the inverse of Tα is defined only for nonvanishing α, cf. Corollary 2.
6. Hamiltonian Formulation In this section we derive the constraints of the DSM. For simplicity we restrict ourselves to closed strings, ∼ = S 1 × R (σ, τ ). Here σ ∼ σ + 2π is the “spatial” variable around the circle S 1 (along the closed string) and τ is the “time” variable governing the Hamiltonian evolution. The discussion will be carried out for more general actions in fact: We may regard any action of the form of SDSM , where D is required to be a maximally isotropic (but possibly non-involutive) subbundle of E; in other words for the present purpose we will consider any action of the form (24) for any orthogonal (X-dependent) matrix Oji . Generalizing an old fact for PSMs [25], the corresponding constraints are “first class” (define a coisotropic submanifold in the phase space), iff D is a Dirac structure (i.e. iff the matrices Oji satisfy the integrability conditions (23) found above). Let ∂ denote the derivative with respect to σ (the τ -derivative will be denoted by an overdot below) and let δ be the exterior differential on phase space. Then we have Theorem 2. For α = 0 and ∼ = S 1 × R, the phase space of SDSM , D maximally ∗ isotropic in E = T M ⊕ T M, may be identified with the cotangent bundle to the loop space in M with the symplectic form twisted by the closed 3-form H , =
δX (σ ) ∧ δpi (σ )dσ − i
S1
1 2
S1
Hij k (X(σ )) ∂X i (σ ) δX j (σ ) ∧ δX k (σ ) dσ , (67)
subject to the constraint Jω,v (σ ) = v i (σ, X(σ ))pi (σ ) + ωi (σ, X(σ ))∂X i (σ ) = 0 , for any choice of ω ⊕ v ∈ C ∞ (S 1 ) ⊗ (X ∗ D), or, in the description of D by means of O ∈ (O(T M)), J ≡ (O + 1)∂X + (O − 1)p = 0 . The constraints are of the first class, if and only if D is a Dirac structure.
(68)
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A. Kotov, P. Schaller, T. Strobl
For Dirac structures D that may be written as the graph of a bivector P (for the splitting chosen—cf. Proposition 1 and Example 1), 1 + O is invertible; then obviously (68) can be rewritten as ∂X − Pp = 0 or ∂Xi + P ij pj = 0 (cf. Eq. (22) and the text about notation following this equation!), which agrees with the well-known expression of the constraints in the WZ-Poisson sigma model [16]. Proof. To derive the Hamiltonian structure we follow the shortcut version of Diracs procedure advocated in [9]. the WZ-term, manipulating For simplicity we first drop − + Ldσ ∧ dτ := SDSM − N3 H in a first step. With dx ∧ dx = −2dσ ∧ dτ we obtain from SDSM by a straightforward calculation α L[X, A± ] = − X˙ 2 + 21 (A+ + αV+ − A− + αV− )X˙ 2 α 2 1 + ∂X + 2 (−A+ − αV+ − A− + αV− )∂X 2 1 1 − 2 V− (A+ + αV+ ) + (A+ V− + A− V+ ) , 4
(69)
˙ and ∂X where appropriate target index contraction is understood [canonically, V , X, carry an upper target-index, and A± (as well as p introduced below) a lower one; but all indices may be raised and lowered by means of the target metric g]. For what concerns the determination of momenta, α = 0 is qualitatively quite different from α = 0. Restricting to the first case for the proof of the present theorem, we may now employ the following substitution to introduce a new momentum field p: α − X˙ 2 + β X˙ 2
∼ →
p X˙ +
1 (p − β)2 , 2α
(70)
1 2 ∼ which results from − α2 X˙ 2 → p X˙ + 2α p after shifting p to p − β. Within an action functional any such two expressions—for arbitrary C-numbers α = 0 and possibly field dependent functions β—are equivalent, classically (eliminate p by its field equations) or on the quantum level (Gaussian path integration over p). Applying this to the first line in (69) with β = 21 (A+ + αV+ − A− + αV− ) and noting that the last bracket in the third line vanishes due to A± ∈ D and the isotropy condition posed on D, we obtain ∼ L→ LHam with
2 1 p − 21 A+ − 21 αV+ + 21 A− − 21 αV− LHam [X, A± , p] = p X˙ + 2α α + ∂X 2 + 21 (−A+ − αV+ − A− + αV− )∂X 2 − 21 V− (A+ + αV+ ) .
(71)
It is straightforward to check that the above terms can be reassembled such that LHam [X, A± , p] = p X˙ − V− p − A− ∂X 2 1 A+ + αV+ − (A− + αV− ) − 2(p + α∂X) + 8α − 21 A− V− − p∂X .
(72)
We now want to show that the last two lines may be dropped in this expression. Here we have to be careful to take into account that A and V are in general not independent fields,
Dirac Sigma Models
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but subject to the restriction that their collection A = A ⊕ V lies in the isotropic subbundle D. First we note that A− V− ≡ 21 A− , A− = 0 due to A− ∈ D. Next, with a shift
+ := A+ + A− , the A− -part drops out in the second line; this is particularly A+ → A obvious in terms of independent fields a± , where the term in the square brackets takes the form [(1 + O) + α(1 − O)] (a+ − a− ) − 2(p + α∂X). After this shift, A− enters the action only linearly anymore, and thus plays the role of a Lagrange multiplier. This already shows the appearance of the constraints Jω,v = 0. Parameterizing ω ⊕ v ∈ D as (1 + O)λ ⊕ (1 − O)λ, one obtains Jω,v = g(Oλ, (O + 1)∂X + (O − 1)p), g being the Riemann metric, which vanishes for any λ (an unconstrained Lagrange multiplier field), iff (68) holds true. To show that the remaining dependence of the lower two lines in (72) on p ⊕ ∂X ∈ E can be eliminated (by a further shift in the fields), is seen most easily in a path-integral type of argument10 : Integrating over A− (i.e. taking the path integral over λ), one obtains a delta function that constrains p ⊕ ∂X to lie in the Dirac structure D; correspondingly, the term p∂X gives no contribution since D is isotropic, and −2(p + α∂X) can be
+ into A¯+ . After these manip + + α V
+ by a further redefinition of A absorbed into A 1 ulations the last two lines reduce to 8α ([(1 + O) + α(1 − O)] a¯ + )2 . This is the only dependence of the resulting action on a¯ + , which thus may be put to zero as well.11 There also exists an argument on the purely classical level for the above consider + ≡ A+ + A− by ation: Denote p ⊕ ∂X, taking values in E, by ψ, and A− and A λD and µD , respectively (the index D so as to stress the restriction to the subbundle D < E). Then LHam = LHam [X, p, λD , µD ] = p X˙ − λD , ψ + f1 (µD − ψ) + f2 (ψ), where f1 and f2 are polynomial functions to be read off from (72) and, as always, ·, · denotes the fiber metric in E. Next we observe that τ (D) = {(v; −Ov)} ∼ = D ∗ has a trivial intersection with D = {(v; Ov)} and E = D ⊕ τ (D); note that τ (D) is also isotropic by construction, but in general will not be a Dirac structure (cf. Eq. (33)); as indicated already by the notation, it can be identified with the dual D ∗ of D by means of ·, · . Thus ψ can be decomposed uniquely into components ψD ∈ D and ∗ ∈ D ∗ , ψ = ψ + ψ ∗ . With ψD µD := µD − ψD the action takes the form LHam = D D ∗ ∗ ∗ the last two contribu˙ p X − λD , ψD + f1 ( µD − ψD ) + f2 (ψ), since for vanishing ψD tions reduce to f1 ( µD ). As a consequence there exists F (ψ, µD ) with values in E such ∗ ) + f (ψ) = F, ψ ∗ . With F = F + F ∗ and due to the isotropy of µ D − ψD that f1 ( 2 D D D ∗ + f ( D ∗ we then obtain LHam = LHam (X, p, µD , λD ) = p X˙ −
λD , ψD 1 µD ), where
λD := λD − FD has been introduced and f1 is a quadratic function in its argument. As before we thus may drop f1 ( µD ) (cf. also footnote 11), obtaining ∼ LHam = LHam [X, p, A+ , A− ] → LHam [X, p, λ] = p X˙ − g( λ, J )
(73)
for some unconstrained Lagrange multiplier field λ ∈ T M. 10
A purely Lagrangian argument will be provided in the subsequent paragraph. Note that on behalf of the permitted values for α the matrix α + 1 + (α − 1)O is non-degenerate—due to Lemma 1 or Corollary 2. So the above statement follows from the field equations of a¯ + , and using a¯ + = 0 in the action is a permitted step in the procedure of [9]. However, even if g has indefinite signature and α = 0 is chosen such that the above quadratic form for a¯ + is degenerate, this contribution can be dropped, since then the action does not depend on directions of a¯ + in the kernel of the matrix, so they also give no contribution to the action (alternatively, a¯ + = 0 may be viewed as a gauge fixation for those directions then). We remark parenthetically that the rank of the matrix may depend on X ∈ M in this case, but dropping the contribution to the action in question obviously is the right step. 11
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A. Kotov, P. Schaller, T. Strobl
Noting that the addition of the Wess-Zumino term only contributes to the symplectic form as specified in (67), we thus proved the main part of the theorem. The statement about the first class property follows from specializing the results of [2], where a Hamiltonian system with symplectic form (67) and currents Jω,v for an arbitrary subset of elements ω ⊕ v ∈ E was considered. The constraint algebra of Jω,v = 0 in the above theorem is an example of the more general current algebra corresponding to an exact Courant algebroid E found in [2]— the first class property is tantamount to requiring a closed constraint or current algebra without anomalies. On the other hand, apparently the action SDSM provides a covariant action functional that produces the above mentioned currents (as constraints or symmetry generators) for the case of an arbitrary Dirac structure. As shown above this is even true if D is maximally isotropic but possibly not a Dirac structure. It is an interesting open problem to consider the Hamiltonian structure of the action (20) for a non-isotropic choice of D (does the more general statement hold true that the functional SDSM defines a topological theory iff D ⊂ E is a Dirac structure?) or likewise to provide some other covariant action functional producing constraints of the form considered in [2] for arbitrary D < E. We already observed above that the discussion of the Hamiltonian structure changes (and in fact becomes somewhat more intricate) for the case of vanishing coupling constant α. E.g. substitutions such as in Eq. (70) are illegitimate in this limit. Moreover, as observed already in previous sections of the paper, the kinetic term Skin is even necessary in general to guarantee the morphism property of the field equations (it becomes superfluous only when D is the graph of a bivector); in fact, the number of independent field equations may even change from α = 0 to α = 0 (with the extreme case of D = T M, where for α = 0 one obtains no equations at all). Thus it is comforting to find ∼ S 1 × R, the Hamiltonian structure of SDSM , D < E Theorem 3. For α = 0 and = maximally isotropic, may be identified with the one found in Theorem 2 above. Proof. For a first orientation we check the statement for D = T M (and H = 0): Then SDSM ≡ 0. The vanishing action S (depending on whatsoever fields) is obviously equiv¯ alent (as a Hamiltonian system) to an action S[X, p, λ] = (pX˙ − λp) (multiplication of the integrand by dσ ∧ dτ here and below is understood). Since this example of D corresponds to O = −1, this latter formulation obviously agrees with what one finds in Theorem 2 for this particular case. This case already illustrates that a transition from S ≡ 0 to S¯ depending on additional fields as written above is an important step in establishing the equivalence. We now turn to the general case, putting H to zero in a first step as in the proof of Theorem 2. Thus we need to analyse S0 [X, A] = Ai ∧ dX i − 21 Ai ∧ V i , with A = A ⊕ V taking values in D. Using the unconstrained field a ≡ a0 dτ + a1 dσ of (53), one finds pX˙ − g(λ, p − (1 + O)a1 ) S¯0 [X, p, a0 , a1 , λ] = −g(a0 , (1 + O−1 )∂X + (O − O−1 )a1 ) . (74) Here the third term just collects all terms proportional to a0 of S0 , as one may show by a straightforward calculation (using the orthogonality of O). The first two terms result ˙ the only appearance of τ -derivatives in S0 . The transition from from g((1 + O)a1 , X), S0 to S¯0 is the obvious generalization of the analogous step from S ≡ 0 to S¯ mentioned
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above and explains the appearance of the new fields p and λ; eliminating these fields one obviously gets back S0 . Next we shift λ according to λ = λ¯ + (1 − O)a0 . This yields ¯ = ¯ p − (1 + O)a1 ) S¯0 [X, p, a0 , a1 , λ] pX˙ − g(λ, −g(a0 , (1 + O−1 )∂X + (1 − O−1 )p) . (75) Note that the last term is already of the form −g(a0 , O−1 J ), with J given by Eq. (68). ¯ can be dropped. We now will argue that the second term, containing the fields a1 and λ, For the case that 1 + O is invertible, this is immediate since then the quadratic form for λ¯ and a1 in (75) is nondegenerate—λ¯ and a1 become completely determined by the remaining fields, without constraining them (cf. also [9]). Otherwise the variation w.r.t. a1 constrains the momentum p, but this constraint is fulfilled automatically by J = 0, resulting from the variation w.r.t. a0 . To turn the last argument into an honest off-shell argument, we perform another shift of variables: With a0 := O−1 λ − 21 λ and a1 := a¯ 1 + 21 O−1 (p − ∂X), Eq. (75) becomes ¯ = ¯ (1 + O)¯a1 ) . S¯0 [X, p, λ, a¯ 1 , λ] p X˙ − g( λ, J ) + g(λ, (76) Now it is completely obvious that the last term, the only appearance of λ¯ and a¯ 1 , can be dropped. This concludes our proof, since the remaining integrand agrees with LHam in (73). As a rather immediate but important consequence the above results imply Theorem 4. The reduced phase space of SDSM (for = S 1 × R and D any maximally isotropic D < E) does not depend on the choice of α ∈ R, the metrics h and g on and M, respectively, or the splitting σ : T M → E. It only depends on the subbundle D. Proof. Independence of α follows from the above two theorems. Independence of h is obvious and likewise the one of g when the constraints are written as Jω,v (σ ) = 0, ∀ω ⊕ v ∈ D, cf. Theorem 2 above. Independence on the choice of the splitting is not so obvious at first sight: the symplectic form (67) depends on H (and not only on its cohomology class), and so implicitly on the splitting, cf. Eq. (28); but likewise do the constraint functions Jω,v (σ ) = ωi ∂X i + v i pi , since the presentation of an element of D < E as ω ⊕ v ∈ T ∗ M ⊕ T M assumes an embedding of T M into E (while T ∗ M < E can be identified canonically as the kernel of ρ ∗ , cf. Eq. (27) as well as our discussion in Sect. 4 above). The coordinate transformation on phase space pi → pi + Bij ∂X j establishes the isomorphism between the two Hamiltonian structures corresponding to different splittings (cf. also [2]). Alternatively we may infer splitting independence also from Propositions 4 and 5 above. Acknowledgement. We are grateful to A. Alekseev for discussions and collaboration on related issues, from which we profited also for the purposes of the present paper.
References 1. Alekseev, A., Kotov, A., Strobl, T.: In preparation 2. Alekseev, A., Strobl, T.: Current algebras and differential geometry. JHEP 0503, 035 (2005)
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3. Alekseev, A., Schaller, P., Strobl, T.: The topological G/G WZW model in the generalized momentum representation. Phys. Rev. D52, 7146–7160 (1995) 4. Bergamin, L.: Generalized complex geometry and the Poisson sigma model. Mod. Phys. Lett. A20, 985–996 (2005) 5. Bojowald, M., Kotov, A., Strobl, T.: Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries. J. Geom. Phys. 54, 400–426 (2005) 6. Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212, 591 (2000) 7. Cattaneo, A.S., Felder, G.: Poisson sigma models and symplectic groupoids. In: Oberwolfach 1999, Quantization of singular symplectic quotients. Progress in Mathematics 198, Basel-Boson: Birkh¨auser, 2001, pp. 61–93 8. Courant, T.J.: Dirac manifolds. Trans. Am. Math. Soc. 319, 631–661 (1990) 9. Faddeev, L.D., Jackiw, R.: Hamiltonian reduction of unconstrained and constrained systems. Phys. Rev. Lett. 60, 1692 (1988) 10. Falceto, F., Gawedzki, K.: Boundary G/G theory and topological Poisson-Lie sigma model. Lett. Math. Phys. 59, 61–79 (2002) 11. Gawedzki, K., Kupiainen, A.: G/H conformal field theory from gauged WZW model. Phys. Lett. B215, 119–123 (1988) 12. Gawedzki, K., Kupiainen, A.: Coset construction from functional integrals. Nucl. Phys. B320, 625 (1989) 13. Gerasimov, A.: Localization in GWZW and Verlinde formula. http://arxiv.org/list/hep-th/9305090, 1993 14. Grumiller, D., Kummer, W., Vassilevich, D.V.: Dilaton gravity in two dimensions. Phys. Rept. 369, 327–430 (2002) 15. Ikeda, N.: Two-dimensional gravity and nonlinear gauge theory. Ann. Phys. 235, 435–464 (1994) 16. Klimcik, C., Strobl, T.: WZW-Poisson manifolds. J. Geom. Phys. 43, 341–344 (2002) 17. Kl¨osch, T., Strobl, T.: Classical and quantum gravity in (1+1)-dimensions. Part 1: A unifying approach. Class. Quant. Grav. 13, 965–984 (1996) 18. Kontsevich, M.: Deformation quantization of Poisson manifolds, I. Lett. Math. Phys. 66, 157–216 (2003) 19. Kosmann-Schwarzbach, Y.: Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory. http://xxx.lanl.gov/abs/math.SG.0310359 20. Kotov, A., Strobl, T.: Covariant curvature and symmetries of Lie algebroid gauge theories. In preparation 21. Lindstrom, U., Minasian, R., Tomasiello, A., Zabzine, M.: Generalized complex manifolds and supersymmetry. Commun. Math. Phys. 257, 235–256 (2005) 22. Liu, Z.-J., Weinstein, A., Xu, P.: Manin triples for Lie bialgebroids. J. Diff. Geom. 45, 547–574 (1997) 23. Mackenzie, K.C.H.: Lie algebroids and Lie pseudoalgebras. J. Algebra 27(2), 97–147 (1995) 24. Park, J.-S.: Topological open p-branes. In: Symplectic geometry and mirror symmetry (Seoul, 2000), River Edge, NJ: World Sci. Publishing, 2001, pp. 311–384 25. Schaller, P., Strobl, T.: Quantization of field theories generalizing gravity Yang- Mills systems on the cylinder. Lect. Notes Phys. 436, 98–122 (1994) 26. Schaller, P., Strobl, T.: Poisson sigma models: A generalization of 2d gravity Yang-Mills systems. In: Finite Dimensional Integrable Systems: Proceedings, Dubna: JINR, 1995. pp. 181–190 27. Schaller, P., Strobl, T.: Poisson structure induced (topological) field theories. Mod. Phys. Lett. A9, 3129–3136 (1994) 28. Severa, P.: Letters to A. Weinstein, unpublished 29. Severa, P., Weinstein, A.: Poisson geometry with a 3-form background. Prog. Theor. Phys. Suppl. 144, 145–154 (2001) 30. Strobl, T.: Gravity in Two Spacetime Dimensions. Habilitationsschrift, RWTH Aachen, 1999 31. Strobl, T.: Talk at ESI-conference on “Poisson geometry”, 2001 32. Strobl, T.: Algebroid Yang-Mills theories. Phys. Rev. Lett. 93, 211601 (2004) 33. Witten, E.: Nonabelian bosonization in two dimensions. Commun. Math. Phys. 92, 455–472 (1984) 34. Witten, E.: Two-dimensional gauge theories revisited. J. Geom. Phys. 9, 303–368 (1992) 35. Zucchini, R.: A sigma model field theoretic realization of Hitchin’s generalized complex geometry. JHEP 0411, 045 (2004) Communicated by N.A. Nekrasov
Commun. Math. Phys. 260, 481–509 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1419-1
Communications in
Mathematical Physics
Precise Coupling Terms in Adiabatic Quantum Evolution: The Generic Case Volker Betz1 , Stefan Teufel2 1 2
Institute for Biomathematics and Biometry, GSF Forschungszentrum, Postfach 1129, 85758 Oberschleißheim, Germany. E-mail: [email protected] Mathematisches Institut, Universit¨at T¨ubingen, Auf der Morgenstelle 10, 72076 T¨ubingen, Germany. E-mail: [email protected]
Received: 30 November 2004 / Accepted: 1 March 2005 Published online: 16 August 2005 – © Springer-Verlag 2005
Abstract: For multi-level time-dependent quantum systems one can construct superadiabatic representations in which the coupling between separated levels is exponentially small in the adiabatic limit. Based on results from [BeTe1 ] for special Hamiltonians we explicitly determine the asymptotic behavior of the exponentially small coupling term for generic two-state systems with real-symmetric Hamiltonian. The superadiabatic coupling term takes a universal form and depends only on the location and the strength of the complex singularities of the adiabatic coupling function. Our proof is based on a new norm which allows to rigorously implement Darboux’ principle, a heuristic guideline widely used in asymptotic analysis. As shown in [BeTe1 ], first order perturbation theory in the superadiabatic representation then allows to describe the time-development of exponentially small adiabatic transitions and thus to rigorously confirm Michael Berry’s [Ber] predictions on the universal form of adiabatic transition histories. 1. Introduction We consider the dynamics of a two-state time-dependent quantum system with state vector ψ ∈ C2 described by the Schr¨odinger equation iε∂t − H (t) ψ(t) = 0 (1) in the adiabatic limit ε → 0. The Hamiltonian H (t), t ∈ R, takes values in the realsymmetric traceless 2 × 2-matrices of the form 1 cos θ (t) sin θ(t) H (t) = 2 . (2) sin θ (t) −cos θ(t) This is the prototype of all adiabatic problems in quantum mechanics. If H (·) ∈ C 1 (R), the system (1) can be decomposed into two scalar equations which are decoupled up
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to errors of order ε. More precisely, when U0 (t) is the orthogonal transformation that diagonalizes the symmetric matrix H (t), then in the basis defined by U0 (t), the adiabatic basis, Eq. (1) reads 0 = iε∂t − H0 (t) ψ0 (t) := U0 (t) iε∂t − H (t) U0 (t)∗ U0 (t)ψ0 (t) (3) with
H0 (t) =
iε 2
1 2
− iε2 θ (t)
θ (t) − 21
ψ0 (t) = U0 (t)ψ(t) =:
and
ψ+ (t) ψ− (t)
.
(4)
We call θ (t) the adiabatic coupling function. The adiabatic subspaces are the subspaces spanned by the adiabatic basis vectors. The statement of the “Adiabatic Theorem” is that each adiabatic subspace is invariant under the dynamics up to errors of order ε, uniformly for finite times. This is not as obvious from (4) as it might seem, because of the ε in front of the time derivative in (3), [BoFo]. The phenomenon of adiabatic decoupling is at the basis of understanding a large number of physical phenomena related to the separation of time scales, ranging from the classical Stern-Gerlach experiment for the measurement of spin to the dynamics of molecules; see [PST, Te] for recent reviews. Adiabatic decoupling is, however, much more subtle than suggested by the classical adiabatic theorem: suppose that H is analytic, limt→±∞ θ (t) = 0 sufficiently quickly and a solution of (3) starts for large negative times in the positive adiabatic subspace, i.e. limt→−∞ |ψ− (t)| = 0. Then according to the adiabatic theorem supt∈R |ψ− (t)| = O(ε), and this bound can not be improved. On the other hand, a more precise analysis shows that limt→∞ |ψ− (t)| = O(e−c/ε ) for some c > 0. Thus, while for finite times the transitions between the adiabatic subspaces are of order ε, they are much smaller in the scattering regime. Despite their exponential smallness, these transitions have important physical consequences such as the non-radiative decay in molecules. Their existence is known for a long time, and the scattering amplitude is described by the Landau-Zener formula, for which rigorous justification has been given in [JKP, Jo]. In the light of the exponentially small scattering amplitude, the adiabatic basis does not seem optimal for describing the solution of the Schr¨odinger equation (1). Instead of using a basis where only in the scattering regime transitions are exponentially small, it is desirable to have a representation where they are exponentially small over the whole time evolution. In order to describe the same scattering amplitudes, such a representation must agree with the adiabatic basis in the scattering regime. This rules out the basis obtained by the true solution of (1) where transitions are zero. The key to finding such a representation are superadiabatic bases. Being natural generalizations of the adiabatic basis, they can be constructed recursively from the adiabatic coupling function θ and have the fundamental property that in the nth superadiabatic basis the off-diagonal elements of the Hamiltonian are of order εn+1 . More precisely, there are a unitary Uεn (t) and functions ρεn (t) = 21 + O(ε 2 ) and cεn (t) such that 0 = iε∂t − Hεn (t) ψ n (t) := Uεn (t) iε∂t − H (t) Uεn (t)∗ Uεn (t)ψ(t) with
Hεn (t)
=
ρεn (t)
ε n+1 cεn (t)
εn+1 c¯εn (t)
−ρεn (t)
and ψ n (t) = Uεn (t)ψ(t) .
(5)
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483
Of course, we have to control the superadiabatic coupling functions cεn (t) in order to learn something about this representation. In general, cεn grows like n!, and thus the sequence ε n+1 cεn (t) diverges for each ε > 0. Consequently, the coupling can not be eliminated completely for fixed ε by going to higher and higher superadiabatic bases. Instead, for each ε > 0 there exists an optimal nε = n(ε) for which n → |ε n+1 cεn | attains its minimum. This defines the optimal superadiabatic basis. The significance of the optimal superadiabatic representation was first noticed by Michael Berry in a work [Ber] subsequently refined by Berry and Lim in [BerLi]. Their results can be translated to our language as follows: Assume that θ is real analytic and at its complex singularities z0 closest to the real axis it has the form θ (z − z0 ) =
−iγ + θr (z) z − z0
(γ ∈ R, z0 ∈ C \ R) ,
(6)
where θr (z) has algebraic singularities of order less than one at z0 . Then near tr = Re(z0 ) and in the optimal superadiabatic representation the off-diagonal elements in (5) are exponentially small and asymptotically Gaussian with superimposed oscillations: πγ − tc − (t−tr )2 r 2ε εnε +1 cεnε (t) = 2i πt e ε e 2εtc cos t−t + R(ε, t), (7) sin 2 ε c where tc = Im(z0 ) and R(ε, t) is of higher order in ε. This form is universal in the sense that it depends only on the first order poles of θ but on no other details. In [BeTe1 ] we showed that when the resulting Schr¨odinger equation is solved using first order perturbation theory, transitions between the superadiabatic subspaces take the universal form of an error function. These transition histories are the objects originally studied by Berry and Lim; they are beautifully numerically illustrated in the adiabatic as well as various superadiabatic bases in [LiBe]. For a discussion about the differences between their general approach and ours we refer to [BeTe1 ]. The arguments of Berry and Lim are non-rigorous; the transition histories in the superadiabatic bases are constructed by a recursive scheme similar to (24)–(27). To this scheme the Darboux principle [Di] is applied, which basically claims that in recursions of the given kind only the highest order poles contribute. Taking this principle for granted, the recursion is solved with θ (t) being just the first term in (6), which is easy and yields (7) after optimal truncation. Despite substantial progress in adiabatic theory during the last decade, e.g. [JoPf, Ne, Sj, Ma], a rigorous justification of Berry’s reasoning remained an open problem until now. A special case has been solved only recently by [HaJo, BeTe1 ]; there 1 1 . − θ (t) = i γ t − tr + itc t − tr − itc This is the simplest case in which θ is real valued on R as it must be, but still the corrections θr to the first order poles are absent. To understand why a result with θr = 0 is not satisfactory and at the same time why a first order pole as leading singularity in θ is the natural choice, we need to go back to the Hamiltonian (2). Obviously, (2) is not the most general choice of H , since it is traceless and its eigenvalues are ±1/2. While it is easy to see that being traceless does not really restrict generality [Ber], constant eigenvalues appear after the transformation to the natural time scale. Berry and Lim [BerLi] observed that the Schr¨odinger equation (s) ψ(s) = 0 (8) iε∂s − H
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V. Betz, S. Teufel
(s), for any traceless real-symmetric Hamiltonian H ˜ ˜ (s) = Z(s) X(s) = ρ(s) cos θ(s) sin θ(s) , H (9) ˜ ˜ X(s) −Z(s) sin θ(s) −cos θ(s) with eigenvalues ρ± (s) = ±ρ(s) = ± X 2 (t) + Z 2 (t) that satisfy ρ+ (s) − ρ− (s) = 2ρ(s) ≥ g > 0 for all s ∈ R, can be brought into the form (1) & (2) through the invertible coordinate transformation
s ρ(u) du . (10) t (s) = 2 0
Additionally and more importantly, they showed that whenever X and Z are in a broad genericity class, the adiabatic coupling function θ (t) in the natural time scale has the form (6) at its singularities. In particular, the generic situation where X and Z are analytic functions is included, but usually yields nonzero corrections θr in (6). In the famous Landau-Zener model X(s) = s, Z(s) = δ, θr has a singularity of order 1/3. Thus any result that claims to prove Berry’s conjecture must be able to accommodate such cases. One contribution of the present work is to do just this: for a fairly general form of the correcting contribution θr we prove (7). We use the original idea of Berry to employ the Darboux’ principle of the nearest singularity; as a heuristic tool, this principle is commonly used in asymptotic analysis [Bo, Di]. Roughly speaking it says that in many asymptotic series expansions the late terms are determined by the nearest singularities of the generating function. Moreover, among the singularities at equal distance, only those of the highest order contribute. As an example, in (6) only the first order poles should matter. While in this generality the Darboux principle is just a guideline and certainly not always true, also in cases where it does apply we know of no instance where it has been translated directly into rigorous mathematics. Thus our second main contribution is a new family of norms adapted to Darboux’ principle. In the present application it allows us to quantify the way in which the first order pole in (6) dominates the remainder θr in the asymptotics of the cεn . Given the importance of Darboux’ principle in asymptotic analysis [Bo], our norms might well see further use in that field. A first sign of this is the connection of the present problem with a system of singularly perturbed ODEs as detailed in Remark 6. Another merit of our approach is that, in principle, it can be applied also to more complicated recurrences as arising, e.g., in the context of constructing precise coupling terms between different electronic levels in molecular dynamics, see [BeTe3 ]. Our paper is organized as follows. In Sect. 2 we state our main result in detail. Apart from the precise asymptotics discussed above, it also includes sharp upper bounds to the functions cεnε away from the transition points tr . In addition, we introduce our norm as the central technical tool and compare it with the widely used Cauchy estimates. In Sect. 3 we recall some tools and results from [BeTe1 ] and prove our main theorem, postponing the proofs of the key inequalities to Sect. 4. There we discuss some mapping properties of our norms, which we believe are of independent mathematical interest, and use them to give the proofs omitted in Sect. 3; the proof of one combinatorial lemma is postponed to the Appendix. Finally, in Sect. 5 we discuss several issues concerned with the transformation (10). Using the argument of Berry and Lim [BerLi] as well as Darboux’Theorem for power series (Theorem 5), we show that indeed our main theorem covers a large class of situations including the generic one where X and Z are analytic. As an example, we present the Landau-Zener model. More examples, non-generic cases and other aspects of our result such as the scattering problem are postponed to [BeTe2 ].
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2. Main Results We formulate our results for the system (1) and (2). However, we have to keep in mind that (1) and (2) arise from the physical problem (8) and (9) through the transformation to the natural time scale (10). Therefore, to be physically relevant, the assumptions of our theorem must be satisfied by all θ arising from generic Hamiltonians of the form (9). As we will see in Sect. 5, for such θ the adiabatic coupling function θ is real analytic and at its complex singularities z0 closest to the real axis it has the form −iγ θ (z − z0 ) = + (z − z0 )−αj hj (z − z0 ), z − z0 N
(11)
j =1
where |Imz0 | > 0, γ ∈ R, αj < 1 and hj is analytic in a neighborhood of 0 for j = 1, . . . , N. We will also see that it is necessary to allow −αj ∈ / N for some j . The technical problem arising from this fact is that by removing the leading singularity one does not obtain a function which is analytic in a larger region. As a consequence, it is not sufficient to use standard Cauchy estimates to show that the remainder terms are asymptotically smaller than the contribution from the leading singularity. While (11) is the natural condition arising from the analysis of the transformation (10), we rephrase it in a more convenient way by introducing the following norm: Definition 1. Let tc > 0, α > 0 and I ⊂ R be an interval. For f ∈ C ∞ (I ) we define f (I,α,tc ) := sup sup ∂ k f (t) t∈I k≥0
tc α+k ≤∞
(α + k)
(12)
and
Fα,tc (I ) = f ∈ C ∞ (I ) : f (I,α,tc ) < ∞ . The idea behind this norm is that although it is defined on the real line, it encodes information about f in the complex plane. To illustrate this and to show the connection with (11), we take I = {t} and obtain f ({t},α,tc ) = C < ∞
⇒
|f (k) (t)| ≤ C
(α + k) tc α+k
∀k ∈ N .
(13)
Consequently f ({t},α,tc ) < ∞ for some α > 0 implies that f is analytic at t and that the Taylor series at t converges at least inside the disk Dtc (t) of radius tc . By the Darboux Theorem for power series (Theorem 5), the parameter α controls the strength of the convergence limiting singularities at the boundary of Dtc (t): suppose that the Taylor series has finitely many singularities on ∂Dtc (t), all of them being of the form as in (11) with (z − z0 )−αk , hk analytic near the origin, αk > 0, then Theorem 5 implies f ({t},β,tc ) < ∞
⇔
β ≥ max αk . k
In particular, θ as given in (11) is an element of F1,tc ({tr }) for tc = Im(z0 ) and tr = Re(z0 ), while the second term of (11) is in Fβ,tc ({tr }) with β = maxj αj . This motivates Assumptions 1 and 2 below.
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Remark 1. One might be tempted to think that for functions f that are analytic in Dtc (t) the norm f ({t},α,tc ) is equivalent to f Cauchy ({t},α,tc ) := sup |f (t + z)| |z|
(tc − |z|)α .
(α)
However, standard Cauchy estimates only yield f Cauchy ({t},α,tc ) < ∞
⇒
∃C > 0 : |f (k) (t)| ≤ C
(α + k + 1) tc α+k
∀k ∈ N, (14)
which is larger than (13) by a factor of k + α. There may be ways to improve, but not up n2 to equivalence of the norms: for the elliptic theta function θ3 (z) = ∞ n=0 z , obviously θ3 (0,α,1) < ∞ if and only if α ≥ 1. On the other hand, an elementary estimate shows that θ3 Cauchy < ∞. The reason for the discrepancy is that the Taylor coefficients of 1 (0, 2 ,1)
functions with a dense set of singularities on the boundary of the disk of convergence (as θ3 has) have worse asymptotics than those of functions with isolated singularities. In many problems of asymptotic analysis this lack of preciseness of · Cauchy ({t},α,tc ) plays no role, since the leading singularity is isolated. Then one can subtract the leading singularity and the remainder is analytic on a slightly larger domain. In that case Cauchy estimates applied to the larger domain yield sufficiently small error terms. However, in our case the form (11) of the function near the singularity requires the use of the precise norms f (I,α,tc ) , since subtracting the leading singularity does not increase the domain of analyticity. We now give precise assumptions and formulate our main result. As indicated in the introduction, the leading order contribution to the superadiabatic coupling function arises from the singularities closest to the real axis. Therefore we essentially distinguish two regimes, reflected by Assumptions 1 and 2 below. If t is close to the real part of such a singularity, we determine the exact asymptotic form of cεn (t) under Assumption 2. If t is sufficiently far away from such a point, we provide sharp upper bounds on cεn (t) using Assumption 1. Assumption 1. For a compact interval I and τ ≥ tc > 0 let θ (t) ∈ F1,τ (I ). Assumption 2. For γ , tr , tc ∈ R let θ0 (t) = i γ
1 1 − t − tr + itc t − tr − itc
be the sum of two complex conjugate first order poles located at tr ± itc with residues ∓iγ . On a compact interval I ⊂ [tr − tc , tr + tc ] with tr ∈ I we assume that θ (t) = θ0 (t) + θr (t) with θr (t) ∈ Fα,tc (I )
(15)
for some γ , tc , tr ∈ R, 0 < α < 1. It turns out that under Assumption 2 the optimal superadiabatic basis is given as the nth ε superadiabatic basis where 0 ≤ σε < 2 is such that nε =
tc − 1 + σε ε
is an even integer.
(16)
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487
Our main result is the leading order asymptotics of ε nε +1 cεnε (t) for t ∈ I . For times t that do not belong to an interval satisfying Assumption 2 we establish bounds on ε nε +1 cεnε (t) which are exponentially smaller than the exponentially small leading order terms near the singularities. In our main theorem we do not only control the asymptotic behavior of the Hamiltonian in the optimal superadiabatic basis, but we also obtain constants which are uniform on compact intervals of the other parameters tc , α and γ . This makes the formulation somewhat involved, but is necessary, e.g., for the study of hermitian but not symmetric Hamiltonians and for the study of the scattering regime, [BeTe2 ]. Theorem 1. Let Jtc ⊂ (0, ∞), Jα ⊂ (0, 1) and Jγ ⊂ (0, ∞) be compact intervals. 1. There exists ε0 > 0 and a locally bounded function φ1 : R+ → R+ with φ1 (x) = O(x) as x → 0, such that for all H (t) as in (2) satisfying Assumption 1 with tc ∈ Jtc , for all ε ∈ (0, ε0 ] and all t ∈ I the elements of the superadiabatic Hamiltonian (5) and the unitary Uεnε (t) with nε as in (16) satisfy n ρ ε (t) − 1 ≤ ε2 φ1 θ , (17) ε (I,1,τ ) 2 √ tc τ nε +1 nε (18) cε (t) ≤ ε e− ε (1+ln tc ) φ1 θ (I,1,τ ) , ε and
Uεnε (t) − U0 (t) ≤ εφ1 θ (I,1,τ ) .
(19)
2. Define g(ε, t) = 2i
2ε πtc
sin
πγ 2
e− ε e− tc
(t−tr )2 2εtc
cos
t−tr ε
−
(t−tr )3 3εtc 2
+
σε t tc
.
There exists ε0 > 0 and a locally bounded function φ2 : R+ → R+ with φ2 (x) = O(x) as x → 0, such that for all H (t) as in (2) satisfying Assumption 2 with tc ∈ Jtc , α ∈ Jα , γ ∈ Jγ , for all ε ∈ (0, ε0 ] and all t ∈ I , tc 3 nε +1 nε cε (t) − g(ε, t) ≤ ε 2 −α e− ε φ2 (M), (20) ε where M = max θ (I,1,t ) , θr (I,α,t ) . Furthermore, (17) and (19) hold with c c τ = tc . Remark 2. Inequality (20) shows that in this case the bound (18) is optimal with respect to the dependence on ε. Again, this is only possible since we use the precise norms f (I,α,tc ) instead of standard Cauchy estimates. Remark √ 3. Note that the explicit term in (20) is asymptotically dominant only if |t −tr | = O( ε). Since typically τ > tc in Assumption 2, for all other times t the bound given in (18) is asymptotically smaller than the error term in (20). Remark 4. It was shown in [BeTe1 ] how to derive from Theorem 1, using first order perturbation theory in the optimal superadiabatic basis, the universal transition histories predicted by Berry [Ber].
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For generic analytic Hamiltonians the whole real line can be covered by intervals satisfying either Assumption 1 or Assumption 2. Under additional conditions on the location of the singularities of θ , we can also consider the scattering problem and recover the well known Landau-Zener formulas for the adiabatic transition amplitudes. Then the decay of the exponentially small coupling for large times can come either from θ (I,1,t ) or c from the τ -dependence of the exponent in (18). We postpone a detailed discussion to [BeTe2 ]. 3. Superadiabatic Representations and Optimal Truncation In this section we prove Theorem 1. The mathematical object to control is the Hamiltonian (5) in the superadiabatic representation. This can be achieved by studying superadiabatic projections. In the simple model at hand, our understanding of these projections and their relation to the unitary is rather complete and has been described in [BeTe1 ]. For the convenience of the reader, we give a synopsis here. Let π0 (t) denote the family of spectral projections to the positive eigenvalue of H (t), called the adiabatic projection. Then the nth superadiabatic projection π (n) (t) =
n
ε k πk (t)
(21)
k=0
is the unique family of operators (which are 2 × 2 matrices in our case) with π (n) (t)2 − π (n) (t) = O(εn+1 ) and iε∂t − H (t), π (n) (t) = O(ε n+1 )
(22) (23)
for all t ∈ R. Here, [A, B] denotes the commutator of the matrices A and B. In the following we will occasionally suppress the dependence on t in the notation. The coefficients πk can be constructed recursively by using the basis 0 −1 X= , Y = −2H, Z = −Y /θ , W = 1 1 0 of R2×2 and making the Ansatz πk = xk X + yk Y + zk Z + wk W . It turns out that wk = 0 for all k, while the remaining coefficients fulfill the recursive differential equations i x1 = − θ , 2
y1 = z1 = 0
(24)
and xn = −i(zn−1 − θ yn−1 ),
yn =
n−1
(−xj xn−j + yj yn−j + zj zn−j ),
(25) (26)
j =1 zn = −ixn−1 .
(27)
Precise Coupling Terms in Adiabatic Quantum Evolution
489
In addition, the differential equation yn = −θ zn
(28)
holds for each n ∈ N. In [BeTe1 ] we construct a unitary matrix Uεn (t) which diagonalizes the self-adjoint matrix π (n) (t) and achieves ρεn (t) ε n+1 cεn (t) n∗ n Uε (t) iε∂t − H (t) Uε (t) = iε∂t − εn+1 c¯εn (t) −ρεn (t) with ρεn (t) = 1/2 + O(ε 2 )
and cεn (t) = (xn+1 (t) − zn+1 (t)) (1 + O(ε)).
(29)
In Theorem 3 we will prove that under Assumption 1, 2 (n − 1)! 1 sup |xn (t)| ≤ θ (1) exp(42 θ (1) ) − , τn 2 t∈I 2 (n − 1)! 1 θ sup |zn (t)| ≤ exp(42 θ (1) ) − , (30) (1) n τ 2 t∈I 2 (n − 2)! θ ) − 1 , sup |yn (t)| ≤ exp(42 (1) τn t∈I where θ (1) = θ (I,1,τ ) . Note that the right-hand side above is O(θ (1) ) as θ → 0 uniformly in τ ≥ tc ≥ inf Jtc > 0. In order to prove (19) one just (1) uses (30) in the explicit formulas for Uεn given in Sect. 3 of [BeTe1 ]. Together with the bounds (30) the analysis of Sect. 3 of [BeTe1 ] actually yields a uniform version of (29): there exists a locally bounded function φ with φ(x) = O(x) as x → 0, such that for all H (t) as in (2) satisfying Assumption 1 with tc ∈ Jtc and for all t ∈I , n ρ (t) − 1/2 ≤ ε 2 φ(θ ) (31) ε (1) and
n c (t) − (xn+1 (t) − zn+1 (t)) ≤ ε(|xn+1 (t)| + |zn+1 (t)|) φ(θ ). ε (1)
(32)
Combining (30) and (32), we obtain |cεn (t)| ≤
n! τ n+1
˜ θ ), φ( (I,1,τ )
where φ˜ has the same properties as φ, uniformly in the class of Hamiltonians just discussed. To arrive at (18), we take nε = tc /ε, use Stirling’s formula and analyze the asymptotics of the terms involved. The procedure is performed in detail in [BeTe1 ], and from the calculations there it is again obvious that uniformity in the Hamiltonians is not lost. The only trivial difference is that since we do not truncate at the optimal value τ/ε of n but rather at tc /ε, we obtain in (18) only a factor of exp − tεc (1 + ln tτc ) instead of exp − τε . Thus we have shown part (i) of Theorem 1.
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V. Betz, S. Teufel
As for part (ii), let M be defined as in Theorem 1. In Theorem 4 we will show that there exists a locally bounded function φ1 with φ1 (x) = O(x) as x → 0, such that for all H (t) as in (2) satisfying Assumption 2 with tc ∈ Jtc , α ∈ Jα , γ ∈ Jγ , and for all t ∈I , t − tr −n−1 n−1+α n! n! 2 sin(γ π/2) Re 1 + i φ1 (M), (33) ≤ xn+1 (t) − i n+1 π tc tc tcn+1 provided θ fulfills Assumption 2 and n is even; zn+1 = 0 in that case. Now (32), (33) and optimal truncation show (20), and the proof of Theorem 1 is finished. Remark 5. In [BeTe1 ], we used (28) and converted the nonlinear recursion into the linear but nonlocal recursive integro-differential equation
t −zn+2 = zn + (θ )2 zn + θ θ zn ds. (34) −∞
Since we treated the special case where θr = 0 in Assumption 2, the calculations were rather explicit and we obtained the analogue of Theorem 1 with even better error bounds. In the general situation, there is no way to avoid the nonlinear recursion (but even so, (34) will be useful). As an added bonus of not resorting to (34), all our results are local. Remark 6. As pointed out to us by Vassili Gelfreich, (24)–(27) is connected to the set of singularly perturbed algebraic-differential equations ∂t X(ε, t) = iεZ(ε, t), ∂t Z(ε, t) = iεX(ε, t) − θ (t)Y (ε, t), Y (ε, t) = ε(−X2 (ε, t) + Y 2 (ε, t) + Z 2 (ε, t)) , with the initial condition X(0, t) = θ (t), Y (0, t) = Z(0, t) = 0. Indeed, consider the formal series expansion for X, Y and Z, i.e. X(ε, t) =
∞
ε k xk+1 ,
k=0
with similar expressions for Y and Z. Then (24)–(27) are just the equations for the coefficients of the expansion. This opens the possibility to treat the problem using e.g. Borel summation, but it is not clear to us whether this would be successful. On the other hand, given the connection above, it may well be that our approach, to be presented in the following section, can be used successfully in the theory of singularly perturbed ODE. 4. Solving the Functional Recursion In this section we examine the recursion (24)–(27) and prove, in particular, the estimates (30) and (33). The main ingredient to our proofs is the family of norms from Definition 1. Recall that for tc > 0, α > 0 and a compact interval I we defined f (I,α,tc ) := sup sup ∂ k f (t) t∈I k≥0
tcα+k ≤∞
(α + k)
(35)
Precise Coupling Terms in Adiabatic Quantum Evolution
and
491
Fα,tc (I ) = f ∈ C ∞ (I ) : f (I,α,tc ) < ∞ .
Often tc and I will be fixed, and then we will simply write . (α) and Fα . The following mapping properties of . (α) are crucial. Proposition 1. Let tc > 0 be fixed, α, β > 0 and t ∈ R. Then (α + k) f (I,α,tc ) ∀k ≥ 0. a) sup ∂ k f (t) ≤ tcα+k t∈I b) f (I,α+1,t ) ≤ f (I,α,tc ) . c c) Let B(α, β) = (α) (β)/ (α + β) denote the Beta function. Then f g (I,α+β,tc ) ≤ B(α, β) f (I,α,tc ) g (I,β,tc ) . Proof. a) and b) follow directly from the definitions. Turning to c), for k ≥ 0 we have k k l k (∂ f g)(t) ≤ ∂ f (t) ∂ k−l g(t) l l=0
≤
k f (α) g (β) k α+β+k
tc
l=0
l
(α + l) (β + k − l).
We thus have to investigate the sum in the last line above and relate it to (α + β + k). To do so, we use a nice trick, which is presumably well known. For −1/2 < t < 1/2 let β 1 hβ (t) := (β) . (36) 1−t Then ∂ n hβ = hβ+n . Now consider ∂ k (hα hβ ). Then on the one hand, α+β 1
(α) (β) = B(α, β)hα+β+k . ∂ k (α + β) ∂ k (hα hβ ) =
(α + β) 1−t =hα+β
On the other hand, of course ∂ (hα hβ ) = k
k k l=0
l
l
∂ hα ∂
k−l
hβ =
k k l=0
l
hα+l hβ+k−l .
Now we take t = 0 and use hβ (0) = (β). Then the above calculations give B(α, β) (α + β + k) =
k k l=0
l
(α + l) (β + k − l).
Inserting this in the calculations from the beginning of the proof of d), we find
(α + β + k) k B(α, β) (∂ f g)(t) ≤ f (α) g (β) α+β+k tc
492
V. Betz, S. Teufel
for each k ≥ 0, and consequently f g (α+β) ≤ B(α, β) f (α) g (β) .
By taking g = 1 in c) we arrive at f (I,α+β,tc ) ≤ tcβ
(α) f (I,α,tc ) .
(α + β)
(37)
We will also need the following somewhat more special property of the norms: t Proposition 2. Let s ∈ I and α > 1. If f ∈ Fα,tc (I ), then t → s f (r) dr ∈ Fα−1,tc (I ), and t (α − 1)|t| f (r) dr f (α) . ≤ max , 1 tc s (α−1) In case α > 2 and |t − s| ≤ tc this simplifies to t f (r) dr ≤ (α − 1) f (α) . s
(α−1)
Proof. We have t f (r) dr
∞
s
and for k ≥ 1 , x k sup ∂ f (s) ds k≥1
∞
0
≤ |t − s| f ∞ ≤ |t − s| f (α)
(α) , tcα
(α−1)+k tc tcα+k = sup ∂ k f = f (α) . ∞ (α + k)
(α − 1 + k) k≥0
The claim now follows from the definition of · (α−1) and the fact (α) = (α −1) (α − 1). Remark 7. The intuition behind the norm f (I,α,tc ) is that when it is finite, the function 1 f behaves equally good or better than the function t → (itc +t) α when taking derivatives. The amazing and useful fact stated in Proposition 1 c) is that multiplication not only leaves this property intact, but even furnishes a factor that becomes small when either α or β become large. It is this property that gets all our estimates going. Theorem 2. Suppose that Assumption 1 holds, and write θ := θ < ∞. (1) (I,1,τ ) Then for each n ∈ N, xn (I,n,τ ) zn (I,n,τ )
1 , 2 1 , 2 2 exp(42 θ (1) ) − 1 .
2 ≤ θ (1) exp(42 θ (1) ) − 2 ≤ θ (1) exp(42 θ (1) ) −
yn (I,n,τ ) ≤
1 n−1
(38) (39) (40)
Precise Coupling Terms in Adiabatic Quantum Evolution
493
Remark 8. The Douglas-Adams-constant M = 42 comes out of our proof in a natural way. Numerical calculations suggest that Theorem 2 holds with M = 1, but this is probably much harder to prove. There is also numerical evidence that the asymptotic 3/2 behavior for large θ (1) is not optimal. It appears that exp(M θ (1) ) is still an upper bound, while exp(M θ (1) ) is not. Proof (of Theorem 2). We define Cn and Dn recursively through C1 = θ (1) /2, D1 = 0 and for n even, Cn−1 (41) Cn = θ (1) Cn−1 + (n−1) Dn−1 for n odd, n−1 k=1 B(k, n − k)(Ck Cn−k + Dk Dn−k ) for n even, (42) Dn = 0 for n odd. We now show that for each n ∈ N, xn (I,n,τ ) ≤ Cn , zn (I,n,τ ) ≤ Cn , yn (I,n,τ ) ≤ Dn .
(43) (44) (45)
This is checked directly for n = 1 and n = 2. Suppose it holds for n − 1. If n is even, then xn = 0 so (43) trivially holds, and (27) implies = xn−1 zn (n) = xn−1 (n−1) ≤ Cn−1 = Cn . (n) Proposition 1 c), (26) and the fact that either xn = 0 or zn = 0 at any given n ∈ N yield yn (n) ≤ ≤
n−1
xk xn−k (n) + yk yn−k (n) + zk zn−k (n)
k=1 n−1
B(k, n − k)(Ck Cn−k + Dk Dn−k ).
k=1
If n is odd, it follows from (25) that yn = zn = 0, and + θ yn−1 ≤ xn (n) ≤ zn−1 (n) (n)
θ 1 (1) θ yn−1 ≤ C + Dn−1 . ≤ zn−1 (n−1) + n−1 (n−1) (1) n−1 n−1
This proves (43)–(45). From (41) it follows immediately that θ θ θ (1) (1) (1) Cn = Cn−1 + Dn−1 = Cn−2 + Dn−1 + Dn−2 n−1 n−1 n−2 n−1 n−1 Dj 1 Dj = θ (1) + . = . . . = C1 + θ (1) j 2 j j =2
Thus it is sufficient to control the Dj . We claim:
j =2
(46)
494
V. Betz, S. Teufel
Lemma 1. For M ≥ 42 and all even n ∈ N, 2j j n/2 1 θ (1) M Dn ≤ . n−1 j!
(47)
j =1
The proof of this purely combinatorial fact is somewhat involved and deferred to the appendix. From (47) and (45) it now follows immediately that yn (n) ≤ Dn ≤
2 exp(M θ (1) ) − 1 n−1
for all n ∈ N. Using (43) and (46) we obtain n−1 2 1 1 xn (n) ≤ Cn ≤ θ (1) + (exp(M θ (1) ) − 1) 2 j (j − 1) j =2 2 1 ≤ θ (1) exp(M θ (1) ) − , 2 and the same estimate applies to zn (n) . The proof is finished.
Proposition 1 a) now immediately implies Theorem 3. Suppose that Assumption 1 holds and write θ = θ . (1) (I,1,τ ) Then for each n ∈ N and each t ∈ R, we have 2 (n − 1)! θ ) − exp(42 θ sup |xn (t)| ≤ (1) (1) τn t∈I 2 (n − 1)! θ sup |zn (t)| ≤ exp(42 θ (1) ) − (1) n τ t∈I 2 (n − 2)! θ ) − 1 . sup |yn (t)| ≤ exp(42 (1) τn t∈I
1 , 2 1 , 2
It is interesting that although the inequalities in Theorem 3 were derived using some seemingly rather crude estimates, they are optimal up to constants in the two most important asymptotic regimes: For large n and each θ (1) as well as for small θ (1) and each n the results in [BeTe1 ] are an example that displays exactly the asymptotic behavior predicted by Theorem 3. Surprisingly, the accurate asymptotics of the recursion (24)–(27) under Assumption 2 are not difficult to obtain from the results of [BeTe1 ] once we have the uniform bounds of Theorem 3 and use our norms. Theorem 4. Let Jtc ⊂ (0, ∞), Jα ⊂ (0, 1) and Jγ ⊂ (0, ∞) be compact intervals. There exists a locally bounded function φ1 : R+ → R+ with φ1 (x) = O(x) as x → 0,
Precise Coupling Terms in Adiabatic Quantum Evolution
495
such that for all H (t) as in (2) satisfying Assumption 2 with tc ∈ Jtc , α ∈ Jα , γ ∈ Jγ and all t ∈ I we have (n − 1)! t − tr −n (n − 1)! −1+α Re 1−i n φ1 (M), (48) xn (t) − icγ ≤ tcn tc tcn (n − 1)! t − tr −n (n − 1)! −1+α Im 1−i n φ1 (M), (49) zn (t) + cγ ≤ tcn tc tcn where M = max θ (I,1,t ) , θr (I,α,t ) and cγ = 2 sin(γπ π/2) . c
c
Proof. Let xn,0 , yn,0 and zn,0 be defined via the recursion (25)–(27) started with x1,0 = iθ0 /2. This is the situation where θ just consists of a pair of simple poles, and Theorem 3 of [BeTe1 ] implies (48) and (49) for xn,0 , yn,0 and zn,0 and any α < 1. Uniformity in the parameters γ and tc is not spelled out there, but again it is easy to derive from the estimates given. Let us now write ξn = xn − xn,0 ,
ηn = yn − yn,0
and
ζn = zn − zn,0 .
The proof will be done as soon as we show |ξn | , |ζn | ≤
(n − 1)! −1+α n φ1 (M) tcn
(50)
for some φ with the properties given in the theorem, uniformly in the parameters α and tc . Without loss we assume tr = 0, and we write Fk instead of Fk,tc (I ) etc. The main step is Lemma 2. ξn ∈ Fn−1+α and ζn ∈ Fn−1+α for each n ∈ N, and there exists φ : R+ → R+ with φ(x) = O(x) as x → 0 such that ζn (n−1+α) ≤ φ(M) and
ξn (n−1+α) ≤ φ(M)
for all n ∈ N, uniformly in α ∈ Jα , tc ∈ Jtc . Proof (of Lemma 2). We first prove the assertion for the ζn . For odd n, ζn = 0, and there is nothing to prove, so let n be even. Since ζ2 = θr /2, the assertion is true for n = 2. Using (28) along with the recursion, we find t −zn+2 = zn + (θ )2 zn + θ θ zn ds + yn (0) , (51) 0
which is obviously just another way to write (34). We decompose (51) into terms which contribute to zn+2,0 and those that do not, with the result t zn+2 = zn,0 + (θ0 )2 zn,0 + θ0 θ0 zn,0 ds + yn,0 (0) (52) 0 =zn+2,0
+ζn
2
+ (θ ) + (θr )2 )zn,0 (53)
t
t
t +θ θ ζn ds + θ0 θr zn,0 ds + θr θ zn,0 ds + θ yn (0) − θ0 yn,0 (0). (54) 0
ζn + (2θ0 θr 0
0
496
V. Betz, S. Teufel
The terms in (53) and (54) contribute to ζn+2 , and we are going to estimate the . (n+1+α) norm of each of them, using Propositions 1 and 2. Starting with (53), we have ζ ≤ ζ (n−1+α) , (55) (n+1+α) 2 ≤ B(2, n − 1 + α) (θ )2 ζ (n−1+α) (θ ) ζn (n+1+α)
and
(2)
2 1 θ ζ (n−1+α) , ≤ (1) (n + α)(n − 1 + α)
(2θ0 θr + (θr )2 )zn,0
(n+1+α)
(56)
≤ B(1 + α, n) θr (θ0 + θ )(1+α) zn,0 (n)
(α) (n) θ + θ zn,0 . θ ≤ r (α) 0 (1) (1) (n)
(n + α + 1)
(57)
Turning to (54), let us first note that t θ ζn ds ≤ (n − 1 + α) θ ζn (n+α) ≤ θ (1) ζn (n−1+α) . 0
(n−1+α)
Similarly, t θ zn,0 ds r 0
(n−1+α)
≤ (n − 1 + α)B(α, n) θr (α) zn,0 (n) =
and
(α) (n) θ zn,0 r (α) (n)
(n − 1 + α)
t θ zn,0 ds 0
(n)
≤ θ (1) zn,0 (n) .
With these estimates we obtain t 2 1 θ θ ζn (n−1+α) , θ ζ ds ≤ n (1) (n + α)(n + α − 1) 0 (n+1+α) t
(α) (n) θ θ θ zn,0 , θr zn,0 ds ≤ r (α) 0 (1) 0 (n)
(n + 1 + α) 0 (n+1+α) t θ θ zn,0 ds ≤ B(1 + α, n) θr (α) θ (1) zn,0 (n) . r 0
(58) (59) (60)
(n+1+α)
Finally, θ yn (0)
(n+1+α)
tc n−1+α θ (2)
(n + 1 + α)
(n) θ yn (n) , ≤ tc −1+α (1)
(n + 1 + α) ≤ |yn (0)|
(61)
Precise Coupling Terms in Adiabatic Quantum Evolution
497
and θ yn,0 (0) 0
(n+1+α)
≤ tc −1+α
(n) θ yn,0 . 0 (1) (n)
(n + 1 + α)
(62)
Now we collect all the estimates from (55) through (62) and obtain
2 θ
ζn (n−1+α) (n + α)(n − 1 + α) θ (n) r (α) + 2 (α) θ0 (1)
(n + 1 + α) zn,0 + ( (α) + (1 + α)) θ
ζn+2 (n+1+α) ≤ 1 +
(1)
(1)
+
1+α (n)
tc
(n + 1 + α)
(n)
yn (n) + θ0 (1) yn,0 (n) . (1)
θ
This shows ζn+2 ∈ Fn+1+α . The above calculations and the bounds from Theorem 2 now + + imply the existence of a locally bounded φ : R → R with φ(x) = O(x) as function x → 0, such that with M = max{θ (1) , θr (α) } and Q = φ(M) we have ζn+2 (n+1+α) ≤ 1 +
Q
(n) ζn (n−1+α) + Q. n(n − 1)
(n + 1 + α)
Moreover, since lim nβ
n→∞
(n) =1
(n + β)
(63)
(n) ≤ 1 for each n ∈ N and β ≥ 1 (cf. [AbSt], 6.1.46 and for each β > 0 and nβ (n+β) 6.1.47), the sequence ( ζn (n+α−1) )n∈N is bounded by the sequence (an )n∈N defined through
2Q Q an+2 = an 1 + 2 + 1+α n n with a2 = ζ2 (1+α) ≤ θr (α) . an is increasing, and so either (Q + 1)an ≤ Q for all n (then an ≤ Q), or eventually an+2 ≤ an
Q n1+α
≤
an (Q+1) , n1+α
2Q Q + 1 1 + 2 + 1+α n n
and then
≤ an
3(Q + 1) 1+ . n1+α
This shows an+2 ≤ a2
n/2 # k=1
3(Q + 1) 1+ (2k)1+α
≤ θ
r (α) exp
3(Q + 1)
∞ k=1
1 (2k)1+α
,
498
V. Betz, S. Teufel
where the infinite sum is bounded uniformly in α > inf Jα > 0. The last inequality above follows by taking the logarithm of the product above and using ln(1 + |x|) < |x| in the resulting sum. Thus we obtain $ ∞ 1 ζn+2 (n+1+α) ≤ max φ(M), θr (α) exp 3(φ(M) + 1) , (2k)1+α k=1
and the claim for ζn is shown. Turning now to the ξn , (25) implies ξn = −i(ζn−1 − θ0 ηn−1 − θr yn−1 ),
(64)
and (28) gives
ηn−1 = −
t
0
(θr zn−1 + θ0 ζn−1 ) ds − yn−1 (0) + yn−1,0 (0).
The claim now follows in a very similar way as above from Propositions 1 and 2.
(65)
By the lemma and Proposition 1 a), ζn ∞ ≤ ζn (n−1+α)
(n + 1 − α) (n − 1)! ≤ φ(M) tc n+1−α tc n
(n + 1 − α) . tc 1−α (n)
Now (63) implies ζn ∞ ≤ cφ(M)
(n − 1)! −1+α n , tc n
with c=
sup
n∈N,α∈Jα
(n + 1 − α) 1−α < ∞, n tc 1−α (n)
and (49) is shown. The same reasoning applies to ξn , showing (48) and finishing the proof. 5. General Hamiltonians Our main result Theorem 1 is formulated for Hamiltonians (2) with constant eigenvalues satisfying Assumptions 1 or 2. In this section we show that these assumptions are satisfied for a large class of Hamiltonians after transformation to the natural time scale. Let us consider (s) ψ(s) = 0 iε∂s − H (66) for the traceless real-symmetric Hamiltonian ˜ ˜ Z(s) X(s) cos θ(s) sin θ(s) H (s) = = ρ(s) . ˜ ˜ X(s) −Z(s) sin θ(s) −cos θ(s)
(67)
Precise Coupling Terms in Adiabatic Quantum Evolution
499
If X 2 + Z 2 > 0, then for each sr ∈ R the transformation
s τ (s) = 2 ρ 2 (u) du
(68)
sr
takes Eq. (66) with Hamiltonian (67) into Eq. (1) with Hamiltonian (2) with θ = θ˜ ◦τ −1 . Berry and Lim [BerLi] found that under very general conditions on X and Z the singularities of θ have the form of a first order pole plus lower order singularities. Then, as to be explained, by the Darboux’ Principle Assumption 1, resp. Assumption 2, are satisfied pointwise on the real line. More precisely, the nth derivative of θ at t ∈ R behaves like
(n)r −n as n → ∞, where r is the distance from t to the nearest pole; the corrections have derivatives going like (n − α)r −n as n → ∞ for some α > 0, and thus are in F1−α,r . The task is now to make this discussion rigorous, and we start with giving a version of Darboux’ Theorem. While this theorem in various forms is certainly well known [He, Bo, Di], we were unable to find a statement in the precision and generality we need in the literature. The proof given in the Appendix uses Cauchy’s formula and explicit integration near the singularities. This strategy was suggested to us by Vassili Gelfreich. Theorem 5 (Darboux’ Principle). Let f be analytic on DR = {z ∈ C : |z| < R}, and assume that f is analytic also on ∂DR except for finitely many points. Assume that there exists N ∈ N and (zj , αj , gj )j ≤N with the properties: (i) For each j , zj is one of the singularities of f on ∂DR ; (ii) αj ∈ R \ {0, −1, −2, . . .}; (iii) The function gj is analytic in a neighborhood Uj of z = 0; (iv) When z0 is a singularity of f on ∂DR and Az0 := {j : zj = z0 }, then f (z) =
(z − z0 )−αj gj (z − z0 ) on
j ∈Az0
%
(Uj + z0 ) ∩ DR .
(69)
j ∈Az0
Then N f (n) (0) −iπαj gj (0) nαj −1 = e 1 + O n1 . n+α j n!
(αj ) z j =1
(70)
j
In words, Darboux’ Theorem says that when f has finitely many algebraic convergence-limiting singularities, each of those contributes to the growth of the derivatives of f with a term of absolute value |g(0)| (n + α)/R n+α depending on the strength g(0), order α and distance R of the singularity, and a phase depending on its strength and location. It is now clear that any function fulfilling the assumptions of Theorem 5 is in Fα,R ({0}), where α = maxj αj . We have to prove this for the analytic continuation of θ , and for this purpose put the following assumptions on the Hamiltonian H˜ : Assumption XZ. X and Z are meromorphic in an open set U ⊂ C containing some point sr on the real axis. X and Z fulfill ρ 2 (s) := X2 (s) + Z 2 (s) ≥ c > 0
∀s ∈ R ∩ U.
(71)
500 (a)
V. Betz, S. Teufel (b)
(c)
Fig. 1. The boundaries of the sets Cr for various situations; the black line is the boundary of CR . In (a) and (b), ρ 2 = 1 + s 2 , and sr = 0 in (a) while sr = 0.5 in (b). The branch cuts are on the imaginary axis. In (c), ρ 2 = (1 + s 2 )2 and sr = 0; examples (b) and (c) show why it is necessary to consider the connected components
By convention, we do not lift removable singularities of ρ 2 , so the critical points of ρ 2 consist of its zeros and the poles of X and Z. For s close to such a critical point s0 of ρ 2 , we require & ' X(s) = (s − s0 )m f (s − s0 ) 1 + (s − s0 )n gX (s − s0 ) (72) & ' Z(s) = ±i(s − s0 )m f (s − s0 ) 1 + (s − s0 )n gZ (s − s0 ) , > −1; the functions f , gX and gZ are analytic in where 0 < n ∈ N, m ∈ Z and 2m+n 2 a neighborhood of s0 , and K := |f 2 (0)(gX (0) − gZ (0))| > 0. The set of critical points has no accumulation points in U . The class of X and Z fulfilling Assumption XZ is smaller than the universality class considered in [BerLi]. It is not our ambition here to investigate just how large we can take our class while still giving a mathematically rigorous proof; but note that (72) does contain the generic case of analytic X and Z and a simple zero of ρ 2 , i.e. m = 0 and n = 1, along with many others. By (71), the map s → ρ(s)2 is analytic and invertible on U ∩ R, but may have critical points in the complex plane. For each such critical point s0 for which a branch cut Bs0 of ρ 2 is needed, we choose the branch cut such ( that it points away from sr , and define U0 to be the connected component of U \ s0 Bs0 containing sr . Then τ as given in (68) is well-defined and analytic in U0 . We define Cr (sr ) := {s ∈ U0 : |τ (s) − τ (sr )| < r}, denote by Cr,0 the connected component of Cr containing sr and put R = R(sr ) := sup{r > 0 : Cr,0 (sr ) ⊂ U0 }. Figure 1 shows some typical cases. We restrict the discussion to the case where Cr (sr ) hits the boundary of U0 at critical points of ρ 2 as r grows to R. When U is large enough, this is the generic case, provided one has put the branch points Bs0 in a sensible way. We distinguish two cases corresponding to our Assumptions 1 and 2 on θ : Assumption R1. Let ∂CR (sr ) ∩ ∂U0 = {s1 , . . . sk }, where the sj are critical points of ρ2.
Precise Coupling Terms in Adiabatic Quantum Evolution
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Assumption R2. Let sr ∈ R be a point where a Stokes line of τ , i.e. a level line of Re(τ (s)), emanating from a critical point s0 of ρ 2 crosses the real axis. Assume that ∂CR (sr ) ∩ ∂U0 = {s0 , s0 }, which means that τ (s0 ) is closer to τ (sr ) than the τ -image of any other critical point of ρ 2 . Theorem 6. Let H˜ , ρ and θ˜ be defined as in (66) with X and Z fulfilling Assumption XZ. Define θ = θ˜ ◦ τ −1 with τ given by (68), and t0 = τ (s0 ). 1. If Assumption R1 holds, then θ ∈ F1,R ({tr }), i.e. Assumption 1 is fulfilled at tr . 2. Suppose that Assumption R2 holds, and that X and Y are given by (72) at s0 . Then ±n with γ = 2m+n+2 there exists a closed interval I sr such that on I , θ (t) = iγ
1 1 − t − t0 t − t0
where θr ∈ Fα,R (I ) for each α ∈ (0, 1) with α ≥
+ θr (t) ,
2m+n 2m+n+2 .
Proof. Let Assumption R1 hold; without loss we assume sr = 0. Then τ is analytic on CR , while on ∂CR there are finitely many singularities. Let s0 be a singularity, and let A0 be the class of functions h which are analytic in a neighborhood of 0 with h(0) = 0. Then for s close to s0 , ρ 2 (s) = 2K(s − s0 )2m+n (1 + h1 (s − s0 )) with h1 ∈ A0 . Consequently
τ (s) − τ (s0 ) = 2
s
(r − s0 )(2m+n)/2 2K(1 + h1 (r − s0 )) dr
s0
√ 2m+n+2 4 2K (s − s0 ) 2 (1 + h2 (s − s0 )) = 2m + n + 2
(73)
with h2 ∈ A0 . Since by construction τ has no critical points inside CR , it is locally analytically invertible there. Since DR := τ (CR ) is the disc with radius R and center τ (sr ), global invertibility follows. Thus τ is one-to-one from CR onto DR with analytic inverse. We have θ˜ (s) 1 d X X Z − Z X θ (τ (s)) = = arctan (s) = (s), 2ρ(s) 2ρ(s) ds Z 2ρ 3
and taking s = τ −1 (t), t ∈ DR , shows that θ is analytic on the circle DR , fulfilling the first assumption of Darboux’ theorem. Note in particular that X Z − Z X is non-singular on DR by our convention of not lifting removable singularities of ρ 2 . For the behavior of θ near a singularity s0 at the boundary of DR , we revisit the calculation of Berry and Lim [BerLi], paying special attention to the error terms that arise. We have (X Z − Z X)(s) = ±inK(s − s0 )2m+n−1 1 + h3 (s − s0 ) ,
502
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with h3 ∈ A0 , A0 as above. Writing σ = s − s0 we now obtain ±inKσ 2m+n−1 (1 + h3 (σ )) ±in(1 + h3 (σ )) = √ 2m+n 3/2 3/2 2(2Kσ ) (1 + h1 (σ )) 4 2Kσ (2m+n+2)/2 (1 + h1 (σ ))3/2 (1 + h3 (σ ))(1 + h2 (σ )) ±in = (2m + n + 2)(τ (s) − τ (s0 )) (1 + h1 (σ ))3/2 ±in (1 + h4 (σ )), = (2m + n + 2)(τ (s) − τ (s0 ))
θ (τ (s)) =
with h4 ∈ A0 , where we used (73) in the second line. It remains to find the form of τ −1 near the singularity τ (s0 ). First note that since τ is an integral, τ (s) − τ (s0 ) = τ (σ ). ˜ α (1 + h2 (σ )) with the obvious K˜ and α. The function From (73), τ (σ ) = Kσ σ → σ K˜ 1/α (1 + h2 (σ ))1/α is invertible in a neighborhood of σ = 0, and the inverse function h5 (σ ) is an element of A0 . We then find 2m+n+2 2j h4 (σ ) = h4 ◦ h5 τ (σ )2/(2m+n+2) = τ (σ ) 2m+n+2 gj (τ (σ )) j =1
with analytic functions gj . Putting things together and writing t = τ (s), t0 = τ (s0 ) we obtain 2m+n+2 2j ±in 1 + θ (t) = (t − t0 ) 2m+n+2 gj (t − t0 ) (74) (2m + n + 2)(t − t0 ) j =1
in a neighborhood of t0 . This has exactly the form (69), and thus Darboux’ Theorem shows (i). As for (ii), note that when Assumption R2 is fulfilled, by continuity of τ still ∂CR (s) ∩ ∂U0 = {s0 , s0 } for all s in a real neighborhood of sr . All the above calculations only need information from the singularities, so they are valid without change, and the proof is finished. Example 1 (Landau Zener√ transitions). In the Landau-Zener model, X(s) = s, Z(s) = δ and consequently ρ(s) = δ 2 + s 2 . The critical points of ρ are at ±iδ, and
δ π δ 3/2 . δ 2 − s 2 ds = R(0) = τ (iδ) = 2 2 0 Moreover, at s = ±iδ the functions X and Z have the form (72) with m = 0, n = 1, f = ±iδ, gX = ∓i/δ and gZ = 0. Thus i 1 + (t ∓ iδ)2/3 g1 (t ∓ iδ) + (t ∓ iδ)4/3 g2 (t ∓ iδ) , θ (t) = ± 3(t ∓ iδ) where g1 and g2 are analytic near 0. Thus θ0 :=
i i − 3(t − iδ) 3(t + iδ)
and θr = θ − θ0 ∈ F1/3,δ (I ) for some I ⊃ {0}. In the simple situation at hand, it is easy to see that in fact θr ∈ F1/3,δ (I ) for every finite interval I . When ρ 2 has more than one critical point on each side of the real axis, the situation is more involved and we refer to [BeTe2 ].
Precise Coupling Terms in Adiabatic Quantum Evolution
503
Appendix Proof of Lemma 1 . We start by converting (41) and (42) to a recursion for the Dn alone by plugging (46) into (42). The result, in a somewhat expanded form, is 2 n−1 θ (1) B(k, n − k) (75) Dn = 4 k=1 2 n−1 k−1 n−k−1 θ Dj Dj (1) B(k, n − k) + + (76) 2 j j j =1
k=2
2 + θ (1)
n−1
B(k, n − k)
j =1
k=2
+
n−2
k−1
Dj j
j =1
n−k−1 l=1
Dl l
(77)
B(k, n − k)Dk Dn−k .
(78)
k=2
To show (47), we will of course proceed inductively. Direct calculation yields that (47) is true up to n = 10 (even for M = 1). Let us now suppose that n ∈ N is an even number and that (47) holds up to n − 2. We will show that (47) also holds for n and for this purpose treat each line of (75) through (78) separately. We start with (78). Using the induction hypothesis, we get k/2 (n−k)/2 n−2 θ 2j M j θ 2l M l (1) (1) 1 1 B(k, n − k) k−1 (78) ≤ j! n−k−1 l! j =1
k=2
=
1 (n−1)(n−2)
n−2
l=1
B(k − 1, n − k − 1)
k/2
j θ 2j (1) M
j =1
k=2
(n−k)/2
j!
θ 2l(1) M l l!
= (∗1 ).
l=1
2 We sort this triple sum by powers p of θ (1) , i.e. take p = j + l. The scheme is the following: p = 2 : j = 1 ⇒ l = 1, k = 2, 3, . . . , n − 2, p = 3 : j = 1 ⇒ l = 2, k = 2, 3, . . . , n − 4 j = 2 ⇒ l = 1, k = 4, 5, . . . , n − 2, .. .
2 so for given p ≤ n/2 (which is the highest power of θ (1) that occurs) we have j running from 1 through p − 1, and for this j we have l = p − j and k = 2j, 2j + 1, . . . , n − 2(p − j ). This gives n/2 n−2p+2j p−1 2p 1 1 θ M p (∗1 ) = (n−1)(n−2) B(k − 1, n − k − 1) j !(p−j )! (1) p=2
=
1 (n−1)(n−2)
n/2
j =1
p−1 n−2p
p θ 2p (1) M
p!
p=2
j =1 k=0
k=2j
p j
B(2j + k − 1, n − 2j − k − 1) . j !(p − j )! ψ(n,p)
504
V. Betz, S. Teufel
We will prove that ψ(n, p) is bounded uniformly in n and p ≤ n/2. We use the integral representation
1 um−1 (1 − u)n−1 du (79) B(n, m) = 0
of the Beta function, and obtain
1 p−1 n−2p p ψ(n, p) = u2j +k−2 (1 − u)n−2j −k−2 du j 0 j =1 k=0
1 p−1 n−2p p = u2j −2 (1 − u)2(p−j )−2 uk (1 − u)n−2p−k du. j 0 j =1
k=0
The sum in the second bracket above is bounded by 2 uniformly on [0, 1], and thus ψ(n, p) is shown to be bounded provided we are able to prove
1 p−1 p
0 j =1
j
u2j −2 (1 − u)2(p−j )−2 du ≤ 5.
(80)
To see (80), first note that the terms with j = 1 and j = p − 1 are equal to p/(2p − 3) and thus bounded by 2 since p ≥ 2. For the remaining terms, we use u2j −2 ≤ uj and (1 − u)2(p−j )−2 ≤ (1 − u)p−j .We then obtain p−2 j =2
p p p 2j −2 2(p−j )−2 u uj (1 − u)p−j = 1 (1 − u) ≤ j j j =0
by the binomial theorem. This proves (80), and we obtain n/2 2p n/2 2p θ (1) M p θ (1) M p 5 1 (78) ≤ ≤ (n − 1)(n − 2) p! 2(n − 1) p! p=2
(81)
p=2
since n ≥ 12. We now turn to (77), and start by proving Lemma 3. For m < n we have
2l m/2 l 1 (m − 2l + 1) θ (1) M ≤ . j m (2l − 1)l!
m Dj j =1
(82)
l=1
Proof. We use the induction hypothesis to calculate 2l 2l j/2 m/2 m m m θ (1) M l θ (1) M l Dj 1 1 . ≤ = j (j − 1) j (j − 1) j l! l! j =1
j =2
l=1
l=1
j =2l
The claim now follows from m k=j
1 1 m−j +1 1 = − = . k(k − 1) j −1 m m(j − 1)
(83)
Precise Coupling Terms in Adiabatic Quantum Evolution
505
Using (82) we get )
n−1
2 (77) ≤ θ
(1) k=2 )
*
j =1
2l (n − k − 2l) θ (1) M l
l=1
(2l − 1)l!
n−k−1 2
×
*
2j (k − 2j ) θ (1) M j B(k, n − k) (k − 1)(n − k − 1) (2j − 1)j ! k−1 2
=: (∗2 ).
2 2 Again we sort this by powers p of θ (1) , leaving the leading θ (1) out. The scheme is p = 2 : j = 1 ⇒ l = 1, k = 3, 4, . . . , n − 3, p = 3 : j = 1 ⇒ l = 2, k = 3, 4, . . . , n − 5, j = 2 ⇒ l = 1, k = 5, 6, . . . , n − 3, .. . and this time the general term is j = 1, . . . , p − 1, l = p − j , k = 2j + 1, . . . , n − B(k,n−k) 2(p − j ) − 1. We use (k−1)(n−k−1) = B(k−1,n−k−1) (n−1)(n−2) and obtain 2p 2 n/2−1 θ θ (1) M p (1) φ(n, p) (84) (∗2 ) = (n − 1)(n − 2) p! p=2
with φ(n, p) =
p−1 n−2p+2j −1 j =1
=
j =1
=
k=2j +1
p−1 n−2p−1 k=1
p j
p j
(k − 2j )(n − k − 2(p − j )) B(k − 1, n − k − 1) (2j − 1)(2(p − j ) − 1)
k(n − k − 2p) B(2j + k − 1, n − 2j − k − 1) (2j − 1)(2(p − j ) − 1)
1 p p u2j −1 (1−u)2(p−j )−1 0 j =1
j
n−2p−1
k(n−2p−k)uk−1 (1 − u)n−2p−k−1 du.
(2j −1)(2(p−j )−1)
k=1
k−1 ≤ 4(n−2p−1) The second sum above is obviously bounded by (n−2p−1) ∞ k=1 ku on [0, 1/2], and since it is symmetric around u = 1/2, this bound is also valid on [0, 1]. On the other hand, the integral representation of the Beta function yields
p 1 p
0 j =1
=
j
p−1 p u2j −1 (1 − u)2(p−j )−1 B(2j, 2(p − j ) du = j (2j − 1)(2(p − j ) − 1) (2j − 1)(2(p − j ) − 1)
1 (2p − 1)(2p − 2)
1 = (2p − 1)(2p − 2)
p−1 j =1
j =1
p B(2j − 1, 2(p − j − 1)) j
1 p−1 p
0 j =1
j
u2j −2 (1 − u)2(p−j )−2 du.
506
V. Betz, S. Teufel
20(n−2p−1) The last integral is bounded by 5 due to (80), and thus φ(n, p) ≤ (2p−1)(2p−2) ≤ Inserting in (84) yields 2(p+1) p+1 n/2−1 θ (1) M 5 (77) ≤ . M(n − 1) (p + 1)!
5(n−2) (p+1) .
(85)
p=2
Turning to (76), note first that for j < n − 1, we have n−1
B(k, n − k) = B(j + 1, n − j − 1)
k=j +1
+B(j + 2, n − j − 2) + . . . + B(n − 1, 1) =
n−j −1
B(k, n − k).
k=1
Moreover, B(1, n − 1) = 1/(n − 1) and B(k, n − k) ≤ 2/((n − 1)(n − 2)) for 2 ≤ k ≤ n − 2, and thus n−1
B(k, n − k) ≤ 4/(n − 1).
k=1
By symmetry, n−1 n−2 k−1 n−1 2 2 Dj Dj (76) = 2 θ (1) B(k, n − k) B(k, n − k) = 2 θ (1) j j k=2 j =1 j =1 k=j +1 2l 2 2 n/2−1 n−2 (n − 2l) θ (1) M l 8 θ (1) 8 θ (1) Dj ≤ , ≤ n−1 j (n − 1)(n − 2) (2l − 1)l! j =1
l=1
where the last inequality is (82). Thus n/2 2p θ (1) M p 16 (76) ≤ . M(n − 1) p!
(86)
p=2
Finally 2 n−1 θ 2 1 (1) (75) = θ (1) B(k, n − k) ≤ . 4 n−1
(87)
k=1
Combining (81), (85), (86) and (87) we arrive at n/2 θ 4 M 2 θ 2p M p 1 1 1 5 21 2 (1) (1) θ + . Dn ≤ + + + (1) n−1 2 M 2! 2 M p! p=3
Choosing M ≥ 42, the proof of Lemma 1 is finished.
Precise Coupling Terms in Adiabatic Quantum Evolution
507
+ Proof of Theorem 5. Let z0 be a singularity of f and write Uz0 = j ∈Az (Uj ). From 0 (69) it is clear that the function z → j ∈Az (z−z0 )−αj gj (z−z0 ) is the unique analytic 0 continuation of f from (Uz0 + z0 ) ∩ DR to (Uz0 + z0 ) \ Bz0 , where Bz0 := {z ∈ C : z = az0 , ( a > 1} is the branch cut (if necessary). Moreover f is analytic on the closed set DR \ j (Uj + zj ), and therefore analytic in a neighborhood of that set. Putting this continuation together with the continuations near each singularity,( we conclude that there exists δ > 0 such that the analytic continuation of f to DR+2δ \ j Bj exists and ( is bounded on (∂DR+δ ) \ j Bj . Here we may choose δ < 1 and sufficiently small to ( guarantee Dδ ∈ Uj and DδR ∈ Uj for all j . Let = ∂DR+δ ∪ N j =1 Cj be the piecewise smooth path that encircles 0 anticlockwise along the boundary of the disk with radius R + δ and avoids the branch cuts Bz0 by encircling clockwise the singularities at zj with a circle Cj of radius δ. Then f (n) (0) 1 = n! 2π i
,
N
f (z) f (z) 1 dz + dz . zn+1 2π i zn+1
f (z) 1 dz = zn+1 2πi
j =1 C
∂DR+δ
j
The first integral is easily estimated through
1 f (z) sup|z|=R+δ |f (z)| dz ≤ = R −n O(n−k ) 2π i n+1 (R + δ)n ∂DR+δ z
∀k ∈ N.
For the contribution of the poles, first note that by (69) we may treat each j ≤ N separately, even if they belong to the same pole. If αj ∈ N, a straightforward computation shows that αj −1 f (z) αj −1 gj (0) n = (−1) 1 + O n1 . Resz=zj n+α n+1 j z
(αj ) z j Noting that Cj is negatively oriented, we conclude that the contribution from poles is the one claimed in (70). k For the remaining terms let gj (z − zj ) = ∞ k=0 bk (z − zj ) . Then 1 2π i
∞
k=0
Cj
f (z) 1 dz = bk n+1 z 2πi
Cj
=
1 2πi
∞
(z − zj )k−αj dz zn+1
k−αj −n
b k zj
k=0
(ζ − 1)k−αj dζ , ζ n+1
(88)
Cj /zj
where we substituted z = zj ζ . The remaining integral is shifted to the origin and can then be solved explicitly in terms of the hypergeometric function F ,
(ζ − 1)k−αj ζ k−αj dζ = dζ n+1 ζ (ζ + 1)n+1 Cj /zj −1
Cj /zj
F =
1 + n, k − αj + 1 ; −δ k − αj + 2 k − αj + 1
zk−αj +1
δ+0i δ−0i
508
V. Betz, S. Teufel
F =
1 + n, k − αj + 1 ; −δ k − αj + 2 k − αj + 1
δ k−αj +1 1 − e−2πiαj .
In the second line above we used the power series expansion ∞ 1 j n+j , = (−ζ ) j (1 + ζ )n+1 j =0
valid for |ζ | = δ < 1. Note also that the branch cut was moved to the positive real axis through the two changes of variables. For k ≤ max(a, 2 − a) we use the asymptotic expansion of the hypergeometric function (cf. [AbSt], 15.7.2 ) 1 + n, k − αj + 1 F ; −δ = (δn)−(k−αj +1) (k − αj + 2) 1 + O n1 , k − αj + 2 which shows that from the first max(a, 2 − a) + 1 terms in each sum only the k = 0 terms contribute to the leading order in (70) with b0
nαj −1 (2 − αj ) e−iπαj sin((1 − αj )π ) nαj −1 e−iπαj = g . (0) j n+α n+α π z j 1 − αj z j (αj ) j
j
As to be shown, for k > max(a, 2 − a) we have k − αj + 1 k − αj + 1 1 + n, k − αj + 1 F =O . ; −δ = O k−αj k − αj + 2 n2−αj n1+ 2
(89)
Hence, these terms do no contribute to the leading order in (70). Note that the sum over k in (88) converges since Dδ|zj | ∈ Uj . To check (89) we use the integral representation of the hypergeometric function,
1
(2 − αj + k) t k−αj 1 + n, k − αj + 1 F dt ; −δ = k − αj + 2
(1 − αj + k) (1) 0 (1 + δt)n+1 s
1 t k−αj t k−αj = (k − αj + 1) dt + dt (1 + δt)n+1 (1 + δt)n+1 s 0 1 s 1 1 s k−αj 1 k−αj +1 ≤ (k − αj + 1) − + t s nδ (1 + δt)n 0 (1 + δs)n+1 k − αj + 1 ≤ (k − αj + 1) s= √1n
=
1 s k−αj + nδ (1 + δs)n+1
(k − αj + 1) δ
1 n1+
k−αj 2
+
1
(1 +
√ δ n n+1 n )
=O
k − αj + 1 n1+
k−αj 2
.
Acknowledgements. We are grateful to Vassili Gelfreich for several helpful remarks. We also profited from discussions with Gero Friesecke, Alain Joye and Florian Theil. V.B. thanks the Mathematics Institute of the University of Warwick for hospitality and the Symposium “Mathematics of Quantum Systems” organized by G. Friesecke for financial support.
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Abramowitz, M., Stegun, I.A.(Eds.): Handbook of Mathematical Functions. 9th printing, New York: Dover, 1972 Berry, M.V.: Histories of adiabatic quantum transitions. Proc. R. Soc. Lond. A 429, 61–72 (1990) Berry, M.V., Lim, R.: Universal transition prefactors derived by superadiabatic renormalization. J. Phys. A 26, 4737–4747 (1993) Betz, V., Teufel, S.: Precise coupling terms in adiabatic quantum evolution. Annales Henri Poincar´e 16, 217–246 (2005) Betz, V., Teufel, S.: Precise coupling terms in adiabatic quantum evolution: extensions and examples. In preparation Betz, V., Teufel, S.: Adiabatic transition histories for Born-Oppenheimer type models. In preparation Born, M., Fock, V.: Beweis des Adiabatensatzes. Zeits. f¨ur Phys. 51, 165–169 (1928) Boyd, J.P.: The Devil’s Invention: Asymptotics, Superasymptotic and Hyperasymptotic Series. Acta Appl. 56, 1–98 (1999) Dingle, R.B.: Asymptotic Expansions: Their Derivation and Interpretation. NewYork-London: Academic Press, 1973 Hagedorn, G., Joye, A.: Time development of exponentially small non-adiabatic transitions. Commun. Math. Phys. 250, 393–413 (2004) Henrici, P.: Applied and computational analysis, Vol. 2. New York: Wiley, 1977 Joye, A.: Non-trivial prefactors in adiabatic transition probabilities induced by high order complex degeneracies. J. Phys. A 26, 6517–6540 (1993) Joye, A., Kunz, H., Pfister, C.-E.: Exponential decay and geometric aspect of transition probabilities in the adiabatic limit. Ann. Phys. 208, 299 (1991) Joye, A., Pfister, C.-E.: Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum. J. Math. Phys. 34, 454–479 (1993) Martinez, A.: Precise exponential estimates in adiabatic theory. J. Math. Phys. 35, 3889–3915 (1994) Lim, R., Berry, M.V.: Superadiabatic tracking of quantum evolution. J. Phys. A 24, 3255–3264 (1991) Nenciu, G.: Linear adiabatic theory. Exponential estimates. Commun. Math. Phys. 152, 479– 496 (1993) Panati, G., Spohn, H., Teufel, S.: Space-adiabatic perturbation theory. Adv. Theor. Math. Phys. 7, 145–204 (2003) Sj¨ostrand, J.: Projecteurs adiabatiques du point de vue pseudodiff´erentiel. C. R. Acad. Sci. Paris S´er. I Math. 317, 217–220 (1993) Teufel, S.: Adiabatic perturbation theory in quantum dynamics. Springer Lecture Notes in Mathematics 1821, Berlin-Heidelberg-New York: Springer, 2003
Communicated by B. Simon
Commun. Math. Phys. 260, 511–525 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1417-3
Communications in
Mathematical Physics
On a Class of Representations of the Yangian and Moduli Space of Monopoles A. Gerasimov1,2,3 , S. Kharchev1 , D. Lebedev1,4 , S. Oblezin1 1
Institute for Theoretical & Experimental Physics, 117259 Moscow, Russia. E-mail: [email protected]; [email protected] 2 Department of Pure and Applied Mathematics, Trinity College, Dublin 2, Ireland 3 Hamilton Mathematics Institute, TCD, Dublin 2, Ireland 4 Max-Planck-Institut fur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. E-mail: [email protected] Received: 25 November 2004 / Accepted: 6 March 2005 Published online: 31 August 2005 – © Springer-Verlag 2005
Abstract: A new class of infinite dimensional representations of the Yangians Y (g) and Y (b) corresponding to a complex semisimple algebra g and its Borel subalgebra b ⊂ g is constructed. It is based on the generalization of the Drinfeld realization of Y (g), g = gl(N ) in terms of quantum minors to the case of an arbitrary semisimple Lie algebra g. The Poisson geometry associated with the constructed representations is described. In particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of G-monopoles defined as the components of the space of based maps of P1 into the generalized flag manifold X = G/B. Thus the constructed representations of the Yangian may be considered as a quantization of the moduli space of the monopoles. 1. Introduction TheYangian Y (g) for a semisimple complex Lie algebra g was introduced by Drinfeld as a certain deformation of U (g[t]) as a Hopf algebra [1–3] (see for recent review [4], and [5]). Recently a construction of the special class of infinite-dimensional representations of Y (gl(N )) and Y (sl(N )) based on the generalization of the Gelfand-Zetlin construction was introduced in [6] (see also [7]). In this paper we generalize this construction to Y (g) for an arbitrary semisimple Lie algebra g. This generalization is based on the proposed generalization of the Drinfeld realization of Y (gl(N )) in terms of quantum minors to the case of Y (g) for an arbitrary semisimple Lie algebra g. We also construct representations of Y (b), where b ⊂ g is a Borel subalgebra of g. One should note that not all of the considered representations of Y (b) may be lifted to the representations of Y (g). We also describe the Poisson geometry behind the constructed representation by defining explicitly the symplectic leaves of the classical versions of the Yangians Y (g) and Y (b). The proposed description of the symplectic leaves reveals a deep connection with the moduli space of G-monopoles, Lie(G) = g. The symplectic leaves as symplectic manifolds turn out to be open parts of the moduli space of monopoles with
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maximal symmetry breaking supplied with the symplectic structure introduced in [8] for G = SU (2), in [9] for G = SU (N ), and in [10] for an arbitrary semisimple Lie group G. Thus our construction of the Yangian representations can be considered as a quantization of the moduli spaces of G-monopoles. The results of this paper support the strong connection between quantum integrable systems and problems of the quantization of various moduli spaces. The demonstrated connection between the Yangian and the quantization of the moduli space of the monopoles is a particular example of this deep relation. We plan to consider its implications to the theory of quantum integrable theories elsewhere. Let us remark that the connection between the Atiyah-Hitchin symplectic structure on the moduli space of the SU (2) monopoles [8] with some particular integrable systems was noted previously in [11, 12]. Finally note that the explicit construction of the representations of Y (g) discussed below appears to be similar to the constructions of the representations of a class of elliptic algebras proposed in [13, 14]. Nevertheless, the main result (Theorem 3.1) seems to be new. The plan of the paper is as follows. In Sect. 2 we provide various descriptions of the Y (g) in terms of the generators and relations. The construction of Y (g) in terms of the generators Ai (u), Bi (u), Ci (u) for a general Lie algebra is proposed. In Sect. 3, we describe a particular class of representations of the Y (g) and Y (b) and give an explicit realization of its generators in terms of difference operators. The main result is formulated in Theorem 3.1. In Sect. 4 we discuss the underlying Poisson geometry and provide a description of the corresponding symplectic leaves of the classical counterpart of the Yangian. It appears that there is an isomorphism between the open part of the symplectic leaves for Y (b) and the open part of the moduli space of the G-monopoles with the maximal symmetry breaking. 2. The Various Presentations of Y (g) We start with the definition of the Yangian for a semisimple Lie algebra g due to Drinfeld [1, 2] (see also [16]) in the form given in [17]. Let h ⊂ b ⊂ g be a simple finite dimensional Lie algebra g of rank over C with a Cartan subalgebra h and a Borel subalgebra b. Let a = ||aij ||, i, j = 1, . . . , be the Cartan matrix of g, be the set of vertices of the Dynkin diagram of g, {αi ∈ h∗ , i ∈ } be the set of simple roots and {αi∨ , i ∈ } be the set of the corresponding co-roots (aij = αi∨ (αj )). There exist positive integers d1 , . . . , d such that the matrix ||di aij || is symmetric. Define 2(α ,α ) the invariant bilinear form on h∗ by (αi , αj ) = di aij , then aij = (αii,αij) . It is convenient to define the generators of the Yangian in terms of the generating series Hi (u) , Ei (u), and Fi (u) , i ∈ : Hi (u) = 1 +
∞
Hi u−s−1 , (s)
s=0
Ei (u) =
∞
(s) Ei u−s−1 ,
Fi (u) =
s=0
∞
(2.1) (s) Fi u−s−1
.
s=0 (s)
(s)
Definition 2.1. The Yangian Y (g) is the associative algebra with generators Hi , Ei , (s) Fi , i ∈ ; s = 0, 1, . . . , and the following defining relations: [Hi (u), Hj (v)] = 0 ,
(2.2)
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[Hi (u), Ej (u) − Ej (v)]+ ı (αi , αj ) , 2 u−v [Hi (u), Fj (u) − Fj (v)]+ ı [Hi (u), Fj (v)] = (αi , αj ) , 2 u−v
[Hi (u), Ej (v)] = −
[Ei (u), Fj (v)] = −ı
Hi (u) − Hi (v) δi,j , u−v
ı (Ei (u) − Ei (v))2 (αi , αi ) , 2 u−v ı (Fi (u) − Fi (v))2 [Fi (u), Fi (v)] = (αi , αi ) , 2 u−v [Ei (u), Ej (u) − Ej (v)]+ ı [Ei (u), Ej (v)] = − (αi , αj ) 2 u−v (0) [E , Ej (u) − Ej (v)] − i , u−v [Fi (u), Fj (u) − Fj (v)]+ ı [Fi (u), Fj (v)] = (αi , αj ) u−v 2 (0) [Fi , Fj (u) − Fj (v)] − , u−v i = j, aij = 0;
(2.3)
(2.4)
[Ei (u), Ei (v)] = −
(2.5)
[Ei (uσ (1) ), [Ei (uσ (2) ), . . . , [Ei (uσ (n) ), Ej (v)] . . . ]] = 0 ,
σ ∈Sn
[Fi (uσ (1) ), [Fi (uσ (2) ), . . . , [Fi (uσ (n) ), Fj (v)] . . . ]] = 0 ,
(2.6)
σ ∈Sn
i = j, n = 1 − aij , where [a, b]+ := ab + ba. Denote Y (b) ⊂ Y (g) the subalgebra generated by Hi (u), Ei (u), i ∈ . There is another closely related set of generators of the Yangian which were given by Drinfeld [3] in the case Y (sl( + 1)). Below we propose the generalization of this description to Y (g) for an arbitrary semisimple Lie algebra g. In the sequel we will use k the following convention: fs = 1, for any fs if k ≤ j which helps to write down the s=j
formulas in a compact way. Lemma 2.1. Any generation series Hi (u) of the form (2.1) can be represented in the following form si −a Hi (u) =
s=i r=1
As u − ı2 (αi + rαs , αs )
Ai (u)Ai (u − ı2 (αi , αi ))
, i = 1, . . . , ,
where Ai (u), i = 1, . . . , are formal series Ai (u) = 1 +
∞ s=0
Ai u−s−1 . (s)
(2.7)
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Proof. Expanding the left- and right-hand sides of (2.7) in powers u−1 and equating the coefficients in u−r−1 one can check that (r)
Hi
=
(0)
(r−1)
asi A(r) s + f (Ai , . . . , Ai
).
(2.8)
s=1
Since the Cartan matrix is invertible, the statement follows by the induction procedure. Let us introduce the generating series Ci (u) and Bi (u): 1/2
Bi (u) = di
Ai (u)Ei (u),
1/2
Ci (u) = di
Fi (u)Ai (u) , i ∈ ,
(2.9)
where di = (αi , αi )/2. Due to (2.7), the change of the generators Hi (u), Ei (u), Fi (u) to Ai (u), Ci (u), Bi (u) is invertible. It is convenient to introduce an additional set of series Di (u). For any i ∈ let Di (u) = Ai (u)Hi (u) + Ci (u)A−1 i (u)Bi (u) ,
(2.10)
where Hi (u) are expressed in terms of Ai (u) in (2.7). Straightforward calculations lead to the following statement. Proposition 2.1. For any i = 1, . . . , the series Ai (u), Bi (u), Ci (u), Di (u) satisfy the relations [Ai (u), Aj (v)] = 0 ,
(2.11)
[Ai (u), Bj (v)] = [Ai (u), Cj (v)] = 0, (i = j ) , [Bi (u), Bj (v)] = [Ci (u), Cj (v)] = 0, (aij = 0, i = j ) , [Bi (u), Bi (v)] = [Ci (u), Ci (v)] = 0,
(2.12)
[Bi (u), Cj (v)] = 0, (i = j ) , (u − v)[Ai (u), Bi (v)] = ıdi Bi (u)Ai (v) − Bi (v)Ai (u) ,
(2.13)
(u − v)[Ai (u), Ci (v)] = ıdi Ai (u)Ci (v) − Ai (v)Ci (u) ,
(2.14)
(u − v)[Bi (u), Ci (v)] = ıdi Ai (u)Di (v) − Ai (v)Di (u) ,
(2.15)
(u − v)[Bi (u), Di (v)] = ıdi Bi (u)Di (v) − Bi (v)Di (u) ,
(2.16)
(u − v)[Ci (u), Di (v)] = ıdi Di (u)Ci (v) − Di (v)Ci (u) ,
(2.17)
(u − v)[Ai (u), Di (v)] = ıdi Bi (u)Ci (v) − Bi (v)Ci (u) .
(2.18)
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In addition, the following analog of the quantum determinant relation holds: Ai (u)Di u − ı2 (αi , αi ) − Bi (u)Ci u − ı2 (αi , αi ) =
−a si s=i r=1
As (u − ı2 (αi + rαs , αs )) .
(2.19)
The described set of relations between Ai (u), Bi (u), Ci (u) and Di (u) is a generalization of the Y (sl( + 1)) relations of [18] to the case of Y (g) with an arbitrary simple Lie algebra g. By (2.11), the coefficients of the generating series Ai (u), i = 1, . . . , define a commutative subalgebra of Y (g). This property will be important in the construction of the representations of the Yangian in the next section. 3. Construction of the Representations of Y (g) and Y (b) In this section the explicit construction of the representations of the Yangian in terms of difference operators is given. The explicit description of the representation of the Yangian in terms of difference/differential operators acting in some functional space is based on the choice of a commutative subalgebra. The elements of this subalgebra act in this representation by multiplication on functions. The proposed construction uses the subalgebra generated by Ai (u) as a distinguished commutative subalgebra. However, we start with an explicit description of the resulting representation and then we make some comments how it could be derived starting with the representation of commutative subalgebra generated by Ai (u) and using the commutation relations between Ai (u), Bi (u), and Ci (u) described in Sect. 2. Let us introduce a set of variables {γi,k ; i ∈ ; k = 1, . . . , mi }, where mi are arbitrary positive integer numbers and let M be the space of meromorphic functions in these variables. Let us define the following difference operators acting on M: βi,k = e
ı 2 (αi ,αi )∂γi,k
(3.1)
,
where ∂γi,k := ∂γ∂i,k is a differentiation over γi,k . It is useful to arrange the variables into the set of polynomials of the formal variable u of degrees mi , i ∈ , Pi (u) =
mi
(u − γi,p ) ,
i ∈ .
(3.2)
p=1
Consider the operators si −a Hi (u) = Ri (u)
s=i r=1
Ps u − ı2 (αi + rαs , αs )
Pi (u)Pi u − ı2 (αi , αi )
,
ı γ P − (α + rα , α ) s i,k i s s mi 2 s=i+1 r=1 −1/2 −1 Ei (u) = di , βi,k (u − γi,k ) (γi,k − γi,p )
k=1
(3.3)
−a si
p=k
(3.4)
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−1/2
Fi (u) = −di
mi
Ri γi,k + ı2 (αi , αi )
k=1 i−1 −a si
×
s=1 r=1
Ps (γi,k − ı2 (αi + rαs , αs ) + ı2 (αi , αi ))
(u − γi,k − ı2 (αi , αi )) (γi,k − γi,p )
(3.5) βi,k ,
p=k
i = 1, . . . , , where Ri (u) will be specified below.
Theorem 3.1. (i) For any set of integer numbers {mi } satisfying the condition li := j =1 mj aj i ∈ Z+ , consider the polynomials Ri (u) =
li
(u − νi,k ),
(3.6)
k=1
where {νi,k , i ∈ , k = 1, . . . , li } is a set of arbitrary complex parameters. Then the operators (3.3)–(3.5) considered as a formal power series in u−1 , form a representation of Y (g) in the space M. This representation is parameterized by a choice of {mi } obeying the above restrictions (li ≥ 0) and by arbitrary complex parameters {νi,k }. (ii) Let {mi } be arbitrary integers and Ri (u) be rational functions of the following form: +
li
Ri (u) =
k+ =1
+ (u − νi,k ) +
,
−
li k− =1
(3.7)
− (u − νi,k ) −
± , i ∈ , k = 1, . . . , li± } is a set of arbitrary complex parameters and where {νi,k ± li+ −li− = j =1 mj aj i . Then the operators (3.3)–(3.4) considered as formal power series in u−1 , form a representation of Y (b) in the space M. This representation is ± parameterized by the choice of {mi } and by arbitrary complex parameters {νi,k }.
Proof. To prove the theorem introduce the following difference operators: −a si Ps γi,k − ı2 (αi + rαs , αs ) −1/2 s=i+1 r=1 −1 , βi,k κi,k = di (γi,k − γi,p )
(3.8)
p=k
and −1/2
= −di κi,k
Ri (γik + ı2 (αi , αi ))
i−1 −a si
×
s=1 r=1
Ps (γi,k − ı2 (αi + rαs , αs ) + ı2 (αi , αi )) βi,k , (γi,k − γi,p ) p=k
(3.9)
Class of Representations of the Yangian and Moduli Space of Monopoles
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where Ri (u) are rational functions of u. It is easy to show that the operators γi,k , κi,k , κi,k satisfy the relations
κi,k γj,l − γj,l κi,k = − ı2 (αi , αi )δi,j δk,l κi,k ,
(3.10)
κi,k γj,l − γj l κi,k = ı2 (αi , αi )δij δkl κi,k ,
γi,k − γj,l − ı2 (αi , αj ) κi,k κj,l = γi,k − γj,l + ı2 (αi , αj ) κj,l κi,k , γi,k − γj,l + ı2 (αi , αj ) = κj,l γi,k − γj,l − ı2 (αi , αj ) , (3.11) κi,k κj,l κi,k
] = 0, i = j, [κi,k , κj,l
Ri (γi,k ) = −di−1 κi,k κi,k p=k
(3.12)
Ps γi,k − ı2 (αi + rαs , αs )
si −a s=i r=1
,
(γi,k − γi,p )
γi,k − γi,p − ı2 (αi , αi )
p=k
κi,k κi,k = −di−1
[κi,k , κi,l ] = 0, k = l ,
−a si
Ri γi,k + ı2 (αi , αi )
s=i r=1
p=k
γi,k −γi,p )
Ps γi,k − ı2 (αi +rαs , αs )+ ı2 (αi , αi )
p=k
.
(γi,k −γi,p + ı2 (αi , αi )
(3.13) Now let us define the generators as si −a Hi (u) = Ri (u)
s=i r=1
Ei (u) =
Pi (u)Pi u − ı2 (αi , αi )
mi k=1
Fi (u) =
Ps u − ı2 (αi + rαs , αs )
mi k=1
,
(3.14)
1 κi,k , i ∈ , u − γi,k
(3.15)
1 , i ∈ , u − γi,k
(3.16)
κi,k
and let R(ui ) be rational functions compatible with the expansion (2.1). Then the relations (2.2), (2.3), and (2.5) may be derived by straightforward calculations. If we further restrict Ri (u) to be polynomial functions then the additional relations (2.4) hold. To complete the proof of the theorem one should verify the relations (2.6), which forms in fact the only non-trivial part of the proof. One can see that such a verification is reduced to the following combinatorial lemma.
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satisfy the relations (3.10), (3.11). Then for any n = Lemma 3.1. Let γik , κik , and κik 2, 3, 4 the following formulas holds: [κi,kσ (1) , [. . . , [κi,kσ (n) , κj,l ]] . . . ] σ ∈Sn
=
ı(α , α ) n n−1 i i (aij + s) · κ˜ i,kσ (1) · · · · · κ˜ i,kσ (n) · κj,l , (3.17) 2 σ ∈Sn
s=0
σ ∈Sn
=
[κi,k , [. . . , [κi,k , κj,l ]]...] σ (1) σ (n)
ı(α , α ) n n−1 i i (aij + s) · κ˜ i,k · · · · · κ˜ i,k , (3.18) · κj,l σ (1) σ (n) 2 σ ∈Sn
s=0
where κ˜ i,k : =
1 γi,k − γj,l +
κ˜ i,k : = κi,k
ı (αi ,αj ) 2
κi,k , (3.19)
1 γi,k − γj,l +
ı (αi ,αj ) 2
.
Proof. We outline the proof of the only non-trivial relations (3.17). Thus, we should calculate the following expression for n = 2, 3, 4: Xn = [κi,kσ (1) , [. . . , [κi,kσ (n) , κj,l ]] . . . ]. (3.20) σ ∈Sn
We consider only the case of non-coincident indexes ki = kj for any i = j . The proof ı (αi ,αj ) . 2
of the general case is quite similar. Let ηij = represented as a sum of n! terms of the first type: n ηij
n
For n = 2, 3, 4, Xn may be
(γi,k − γj,m )−1 [κi,kσ (1) , [. . . , [κi,kσ (n) , κj,l ]+ ]+ . . . ]+
(3.21)
k=1
and nn − n! terms of the second type: n−s ηiis ηij
s
(γi,k − γj,m )−1
(γi,α − γi,β )−1
α,β
k=1
×[κi,kσ (1) , [. . . , κj,l [κi,kσ (α) , κi,kσ (β) ]]+ ]+ . . . ]+ .
(3.22)
One can reduce the terms of the second type to ( n2 ) terms of the first type as follows. Given In := {1, . . . , n} and the set of variables {ai , i ∈ I }, d, the following iterative formula holds: (ai − d)−1 = (ar − d)−1 (ai − ar )−1 . i∈In
r∈In
i∈In \{r}
Class of Representations of the Yangian and Moduli Space of Monopoles
The left-hand side
n
519
(as − d)−1 of this formula is exactly of the first type for as := γi,ks
s=1
and d := γj,l . The iterations on the right-hand side coincide with all the terms of the second type (3.22) and thus we are left with only the terms of the first type. The simple transformations then lead to (3.17). Finally, let us briefly explain how these representations naturally arise from the rela(s) tions (2.11)–(2.19). Due to (2.11), Ai , i ∈ , s = 0, 1, 2, . . . , generate a commutative subalgebra of Y (g). We would like to construct the representation in the space (s) of functions of the finite collection of the variables {γik } such that Ai act through the multiplication by certain functions of {γik }. It is natural to look for the representation of Ai (u) in the form Ai (u) = Xi (u)Pi (u), where Pi (u) are given by (3.2) and (s) −s−1 are some γik -independent series. From the commutation Xi (u) = 1 + ∞ s=0 Xi u relations (2.11)–(2.19) one derives that Bi (γi,k ) and (Ci (γi,k ))−1 are proportional to the shift operator (3.1). Therefore, by (2.9) the residues of Ei (u) and Fi (u) are proportional to the Bi (γi,k ) and Ci (γi,k ) respectively. This explains the ansatz (3.15), and (3.16) for the generators Ei (u) and Fi (u). 4. Symplectic Leaves of the Yangian and the Monopole Moduli Spaces In this section we describe the Poisson geometry relevant to the description of the Yangian representations proposed above. It appears that this leads to the direct connection with moduli spaces of G-monopoles such that Lie(G) = g. Let Ycl (g) and Ycl (b) be the Poisson algebras corresponding to the classical limit of Y (g) and Y (b) in the sense of [20, 21]. The elements Ycl (g) may be described as functions on the formal loop group LG− based at the trivial loop g(u) = e, where e ∈ G is unity element. We use its parameterization in terms of the infinite series of the form (s) u−s−1 . F (u) = ∞ F s=0 The description of the Poisson algebra Ycl (g) in terms of the generators and relations can be obtained from (2.2)–(2.6) by taking the limit → 0: {hi (u), hj (v)} = 0 , {hi (u), ej (v)} = − (αi , αj ) {hi (u), fj (v)} = (αi , αj )
(4.1)
hi (u)(ej (u) − ej (v)) , u−v
hi (u)(fj (u) − fj (v)) , u−v
{ei (u), fj (v)} = − δij
hi (u) − hi (v) , u−v
(ei (u) − ei (v))2 , u−v (fi (u) − fi (v))2 {fi (u), fi (v)} = (αi , αi ) , u−v {ei (u), ei (v)} = −(αi , αi )
(0)
ei (u)(ej (u) − ej (v)) {ei , (ej (u) − ej (v))} − , {ei (u), ej (v)} = −(αi , αj ) u−v u−v
(4.2)
(4.3)
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fi (u)(fj (u) − fj (v)) {fi , (fj (u) − fj (v))} − , u−v u−v (4.4) i = j, aij = 0;
{fi (u), fj (v)} = (αi , αj )
{ei (uσ (1) ), {ei (uσ (2) ), . . . , {ei (uσ (n) ), ej (v)} . . . }} = 0 ,
σ ∈Sn
{fi (uσ (1) ), {fi (uσ (2) ), . . . , {fi (uσ (n) ), fj (v)} . . . }} = 0 ,
(4.5)
σ ∈Sn
n = 1 − aij , (i = j ) . There exists the following simple interpretation of the generators hi (u), ei (u), fi (u). Let us fix the Gauss decomposition of an element g(u) ∈ LG− : g(u) = exp
fα (u)Fα · exp
α
· exp
φi (u)Hi
eα (u)Eα
,
(4.6)
α
i=1
where Hi , Fα , Eα provide a basis of g labeled by positive roots α, and φi (u), fα (u), eα (u) are the local exponential coordinates on the group LG− . Note that we consider the functions on the formal loop group LG− (i.e. we deal with functions on the formal neighbourhood of e ∈ LG− ) and thus the Gauss decomposition (4.6) is valid “everywhere”. The functions ei (u) := eαi (u), fi (u) := fαi (u) corresponding to the simple roots αi together with hi (u) = exp{− j =1 aj i φj (u)} give us a set of the generators satisfying the relations (4.1)–(4.5). The local coordinates φi (u), ei (u), fi (u) may be expressed explicitly in terms of the matrix elements of the fundamental representations of U (g). Let {πi } be a set of fundamental representations corresponding to the fundamen(i) tal weights {ωi } of g and v+/− be the highest/lowest vectors in these representations. Denote by ai (u), bi (u), ci (u), di (u) the following formal series: (i)
(i)
ai (u) = v− |πi (g(u))|v+ , (i)
(i)
(i)
(i)
bi (u) = v− |πi (g(u))πi (Fi )|v+ ,
(4.7)
ci (u) = v− |πi (Ei )πi (g(u))|v+ , (i)
(i)
di (u) = v− |πi (Ei )πi (g(u))πi (Fi )|v+ . Lemma 4.1. The coordinates corresponding to the simple roots and Cartan elements entering the Gauss decomposition (4.6) can be expressed through the matrix elements (4.7) as follows: eφi (u) = ai (u) , bi (u) ei (u) = , ai (u) ci (u) fi (u) = . ai (u)
(4.8)
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The variables ai (u), bi (u), ci (u), di (u) are the classical counterparts of the variables Ai (u), Bi (u), Ci (u), Di (u) defined in (2.7), (2.9), (2.10). A similar description of Y (gl( + 1)) was given in [3] (see also [19, 22]). In this case the functions ai (u), bi (u), ci (u), di (u) are given by the minors of the matrix π∗ (g(u)), where π∗ is the tautological representation π∗ : gl( + 1) → End(C+1 ), which agrees with (4.7). Let us remark that (4.7) may be considered as a classical counterpart of the universal R-matrix map of the dual Hopf algebra with the opposite comultiplication A0 to the Hopf algebra A (see [2] for details). Together with the explicit Gauss form of the universal R-matrix this should provide the quantum version of Lemma 4.1. The simplest example of Y (sl(2)) may be extracted from [17]. Now consider the classical counterpart of the Yangian representations constructed in Sect. 3. These representations have the following property: the images of Ei (u) are rational operator-valued functions in u with simple poles. Given the Poisson brackets (4.1)–(4.5) on LG− , the representations of the Yangian correspond to the symplectic leaves in LG− . A similar description holds for Ycl (b) in terms of the symplectic leaves in LB− . We define the symplectic leaf O to be rational if the restriction of the generators ei (u) is a rational function over u and let O(0) ⊂ O be an open part corresponding to ei (u) having only simple poles. Thus the symplectic leaves corresponding to the representations constructed in Sect. 3 are rational. One can describe the symplectic leaves as symplectic manifolds as follows. Open parts O(0) of the rational symplectic leaves in LB− corresponding to the representations constructed in Theorem 3.1 are isomorphic b (m) of the based rational (as abstract manifolds) to the open subsets of the space of M maps e
1 e = (e1 , . . . , e ) : (P1 , ∞) −→ (P · · × P1 , 0 × · · · × 0), × ·
(4.9)
of the fixed multi-degree m = (m1 , . . . , m ), where ei (u) are the generators of Ycl (b) corresponding to simple roots. Analogously open parts O(0) of the rational symplectic leaves of LG− corresponding to the representations constructed in Theorem 3.1 are b (m) with additional restrictions mj aj i = isomorphic to the open subsets of M j =1 li ∈ Z+ . Taking into account the results of [15] one can reformulate the description of the symplectic leaves as follows. Consider the space M(m) of the holomorphic maps P1 → G/B ∨ of multi-degree m = (m1 , . . . , m ) ∈ ∨ W , where W = H2 (G/B, Z) is the co-weight lattice of g. It will be useful to consider G/B as a manifold parameterizing the Borel subgroups in G. Choose some Borel subgroup B+ and let b+ ⊂ G/B be the corresponding point in the flag manifold. Let us fix the local coordinate on P1 and consider the evaluation map ev∞ : M(m) −→ G/B defined as ev∞ : f → f (∞). Thus M(m) is supplied with the structure of the fibred space over G/B and the fibre is naturally identified with the moduli space Mb (m) of the based holomorphic maps f : (P1 , ∞) → (G/B, b+ ) of the multi-degree m. It appears that the open part of Mb (m) can be naturally identified b (m) introduced above. Actually this follows from the results with the moduli space M of Drinfeld in the form presented in [15]. Thus it was shown in [15] that Mb (m) is a smooth manifold of dimension dim Mb (m) = 2|m| = 2(m1 + · · · + m ). The explicit description of the manifold Mb (m) can be obtained by generalizing the classical Pl¨ucker embedding of G/B into the product i⊂ P(Vωi ) of the projectivisations of the fundamental representations Vωi as follows. Let πλ : U (g) → End(Vλ ) be
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the irreducible representation of the universal enveloping algebra U (g) with the highest weights λ and Vλ = Vλ ⊗ OP1 be the corresponding trivial vector bundles on P1 . Using λ the highest weight the local coordinate we identify (A1 , Vλ ) = Vλ ⊗C[u]. Denote by v+ λ vectors in Vλ with respect to the Borel subgroup B+ . Similarly let v− be the lowest weight λ |v λ = 1. We also vector in the dual representation Vλ∨ normalized by the condition v− + λ , π (E )v λ , where F and E are the introduce the additional set of vectors πλ (Fi )v+ λ i − i i λ ∈ (A1 , V ) we generators corresponding to the simple roots αi . Thus given a section v λ i λ + λ λ have the following decomposition: v λ (u) = aλ (u) · v+ i∈ bλ (u)πλ (Fi )v+ + φ (u) λ 1 λ λ λ λ with φ ⊂ (A , Vλ ) satisfying v− |φ = v− |πλ (Ei )|φ = 0. According to Drinfeld (see [15] for the details) the moduli space Mb (m) is isomorphic to the space Zm of the collections of sections v λ (u) for each λ ∈ + W satisfying the conditions: 1. The polynomial aλ (u) is monic of degree m, λ; λ ) is strictly less than m, λ; 2. The degree of (v λ − aλ (u)v+ 3. For any G-equivariant morphism φ : Vλ ⊗ Vµ → Vν such that ν = µ + λ and the ν ) = v λ ⊗ v µ we have φ(v λ ⊗ v µ ) = v ν ; conjugated morphism satisfies φ ∗ (v− − − 4. For any G-equivariant morphism φ : Vλ ⊗ Vµ → Vν such that ν < µ + λ we have φ(v λ ⊗ v µ ) = 0. It is easy to see that the set {v λ , λ ∈ + W } satisfying these conditions is determined by its subset {v ωi } corresponding to fundamental representations ωi . Moreover, given arbitrary polynomials aωi (u) and bωi i (u) satisfying Conditions (1) and (2) above (i.e. aωi (u) are monic and deg(aωi ) = deg(bωi i ) + 1 = mi ) there exist such φ ωi (u) that for ωi ωi v ωi (u) = aωi (u) · v+ + bωi i (u)πωi (Fi )v+ + φ ωi (u) Conditions (3) and (4) hold (see [15, 10] for details). Let us consider the subset of the polynomials aωi and bωi such that the roots γi,k of aωi (u) do not coincide γi,k = γj,l for (i, k) = (j, l). The space of such polynomials ai (u) ≡ aωi (u), bi (u) ≡ bωi i (u) is 2|m|-dimensional and thus is isomorphic to the open subspace in the moduli space Mb (m). Note that the polynomials b (m), ai (u) and bi (u) define the map e ∈ M e
1 (P1 , ∞) −→ (P · · × P1 , 0 × · · · × 0), × ·
given by e(u) = (b1 (u)/a1 (u)) × · · · × (b (u)/a (u)). Therefore we have established the isomorphism of the open parts of the moduli spaces b (m). φ : Mb (m) −→ M One can summarize this in the following Proposition 4.1. (i) The open parts O(0) of the rational symplectic leaves of Ycl (b) corresponding to the representations constructed in Theorem 3.1 are isomorphic to the open parts of the spaces of the based maps (P1 , ∞) → (G/B, b+ ) of the fixed multi-degree m = (m1 , . . . , m ) ∈ H2 (G/B, Z).
(4.10)
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(ii) The open parts O(0) of the rational symplectic leaves of Ycl (g) corresponding to the representations constructed in Theorem 3.1 are isomorphic to the spaces of the based maps (4.10) with additional restrictions j =1 mj aj i = li ∈ Z+ . The connection with the explicit parameterization used in the previous sections is as b (m) by the following e´ tale coorfollows. Let us parameterize the open subset U ⊂ M dinates: (xi,k , yi,k ),
i = 1, . . . , ,
k = 1, . . . , mi ,
defined by the conditions ai (xi,k ) = 0,
yi,k = bi (xi,k ).
Then the coordinates (xik , yik ) are related to the coordinates (γi,k , κi,k ) by the simple redefinition xi,k = γi,k , yi,k = κi,k (γi,k − γi,s ) .
(4.11) (4.12)
s=k
Note that the classical limit of the relations (3.10), (3.11) provides the open part of the space Mb (m) with the holomorphic symplectic structure {γi,k , γj,l } = 0, {γi,k , κj,l } = {κi,k , κj,l } =
1 2 (αi , αi )δi,j δk,l κj,l κi,k κj,l (αi , αj ) , γi,k − γj,l
(4.13) (4.14)
, (i, k) = (j, l),
(4.15)
or equivalently in the coordinates (xi,k , yi,k ), {xi,k , xj,l } = 0, {yi,k , yi,l } = 0,
(4.16) (4.17)
{xi,k , yj,l } = 21 (αi , αi )δi,j δk,l yj,l , yi,k yj,l {yi,k , yj,l } = (αi , αj ) , i = j. xi,k − xj,l
(4.18) (4.19)
In general the symplectic leaves of the Poisson-Lie group G are connected components of the intersections of the double cosets of the Poisson-Lie dual G∗ in G × G with the diagonal G ⊂ G × G [21]. We are going to discuss the connection of this description with the algebro-geometric description considered above in a separate publication. It appears that description of the symplectic leaves of the Poisson-Lie groups associated with the Yangian given in this section provides a direct connection with the moduli spaces of the G-monopoles with the maximal symmetry breaking. These moduli spaces are also given by the spaces of the based maps P1 → G/B [23–25]. Our construction provides the holomorphic symplectic structure on these spaces. The explicit description of the holomorphic symplectic structure on the moduli space of the monopoles was given in the case of G = SU (N ) in [9] (generalizing the results for SU (2) of [8]) and for general case in [10]. It turns out that our description of the coordinates on the moduli space and the expression for the Poisson structure in coordinates (xi,k , yi,k ) exactly matches the description given in [10]. Thus the representation constructed in Sect. 2 can be considered as a quantization of the moduli space of the monopoles.
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Acknowledgement. The authors are grateful to A. Levin and A. Rosly for useful discussions and to M. Finkelberg for the explanation of the results of [10, 15]. The research was partly supported by grants CRDF RM1-2545; INTAS 03-513350; grant NSh 1999.2003.2 for support of scientific schools, and by grants RFBR-040100646 (A. Gerasimov, D. Lebedev), and RFBR-040100642 (S. Kharchev, S. Oblezin). The research of A. Gerasimov was also partly supported by SFI Basic Research Grant. D. Lebedev would like to thank the Max-Planck-Institut f¨ur Mathematik for financial support and hospitality. S. Oblezin is deeply grateful to the Independent University of Moscow for support.
References 1. Drinfeld, V.G.: Hopf algebras and the quantum Yang-Baxter equation. Dokl. Akad. Nauk SSSR 283, 1060–1064 (1985); translation in Soviet Math. Dokl. 32, 254–258 (1985) 2. Drinfeld, V.G.: Quantum groups. In: Proc. Int. Congr. Math. Berkeley, California (1986), Vol. 1, Providence, RI: AMS, 1987, pp. 718–820 3. Drinfeld, V.G.: A new realization of Yangians and of quantum affine algebras (Russian) Dokl. Akad. Nauk SSSR 296, no. 1, 13–17 (1987); translation in Sov. Math. Dokl. 36, 212–216 (1988) 4. Molev, A., Nazarov, M., Olshanski, G.: Yangians and classical Lie algebras. Russ. Math. Surv. 51, 205–282 (1996) 5. Molev, A.: Yangian and their applications. In: Handbook of Algebra, Vol.3, M. Hazewinkel, ed., London-Amsterdam-New York: Elsevier, 2003, pp. 907–959 6. Gerasimov, A., Kharchev, S., Lebedev, D.: Representation theory and quantum inverse scattering method: the open Toda chain and the hyperbolic Sutherland model. Int. Math. Res. Notices 17, 823–854 (2004) 7. Gerasimov, A., Kharchev, S., Lebedev, D.: On a class of integrable systems connected with GL(N, R). Int. J. Mod. Phys. A 19, Suppl., 205–216 (2004) 8. Atiyah, M.F., Hitchin, N.: The geometry and dynamics of magnetic monopoles. Princeton, NJ: Princeton University Press, 1988 9. Bielawski, R.: Asymptotic metrics for SU (N)- monopoles with maximal symmetry breaking. Commun. Math. Phys. 199, 297–325 (1998) 10. Finkelberg, M., Kuznetsov, A., Markarian, N., Mirkovi´c, I.: A note on the symplectic structure on the space of G-monopoles. Commun. Math. Phys. 201, 411–421 (1999) 11. Vaninsky, K.: The Atiyah–Hitchin bracket and the open Toda lattice. J. Geom. and Phys. 46, 283–307 (2003) 12. Vaninsky, K.: The Atiyah-Hitchin bracket for the cubic nonlinear Schrodinger equation. I. General potentials. http://arxv.Org/list/,2001 13. Feigin, B., Odesskii, A.: Vector bundles on elliptic curves and Sklyanin algebras. In: Topics in quantum groups and finite type invariants, Mathematics at the Independent University of Moscow, eds. B.Feigin V.Vasiliev Advances in Mathematical Sciences 38, AMS Translations, ser.2, Vol. 185, Providence RI: Amer. Math. Soc., 1998, pp. 65–84 14. Feigin, B., Odesskii, A.: Elliptic deformations of current algebras and their representations by difference operators. Funct. Anal. Appl. 31, 57–70 (1977) (Russian); translation in Funct. Anal. Appl. 31, 193–203 (1997) 15. Finkelberg, M., Mirkovi´c, I.: Semi-infinte flags. I. Case of global curve P1 . In: Differential topology, infinite-dimensional Lie algebras, and applications. D. B. Fuchs’ 60th anniversary collection eds: A. Astashkevich et al, AMS Translations, Ser. 2, 194, Providence, RI: Amer. Math. Soc., 1999, pp. 81–112 16. Chari, V., Pressley, A.: A guide to quantum groups. Cambridge: Cambridge Univ. Press, 1994 17. Khoroshkin, S.M., Tolstoy, V.N.: Yangian Double. Lett. Math. Phys. 36, 373–402 (1996) 18. Nazarov, M., Tarasov, V.:Yangians and Gelfand-Zetlin bases. Publ. Res. Inst. Math. Sci. 30, 459–478 (1994) 19. Ding, J., Frenkel, I.: Isomorphism of two realizations of quantum affine algebra Uq (gl(n)). Commun. Math. Phys. 156, 277–300 (1993) 20. Drinfeld, V.G.: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classicalYang-Baxter equations. Dokl. Akad. Nauk SSSR 268, no. 2, 285–287 (1983); translation in Sov. Math. Dokl. 27, 68–71 (1983) 21. Semenov-Tian-Shansky, M.A.: Dressing transformations and Poisson group actions. Publ. Res. Inst. Math. Sci. 21(6), 1237–1260 (1985) 22. Iohara, K.: Bosonic representations of Yangian double DY (g) with g = glN , slN . J.Phys. A29, 4593–4621 (1996) 23. Hurtubise, J.: The Classification of the monopoles for the classical groups. Commun. Math. Phys. 120, 613–641 (1989)
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24. Hurtubise, J., Murray, M.K.: On the construction of monopoles for the classical groups. Commun. Math. Phys. 122, 35–89 (1989) 25. Jarvis, S.: Euclidean monopoles and rational maps. Proc. Lond. Math. Soc.(3) 77, 170–192 (1998) Communicated by G.W. Gibbons
Commun. Math. Phys. 260, 527–556 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1420-8
Communications in
Mathematical Physics
Convergence to Equilibrium for Intermittent Symplectic Maps Carlangelo Liverani1 , Marco Martens2 1 2
Dipartimento di Matematica, II Universit`a di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma, Italy. E-mail: [email protected] University of Groningen, Department of Mathematics, P.O. Box 800, 9700 AV Groningen, The Netherlands. E-mail: [email protected]
Received: 1 December 2004 / Accepted: 23 February 2005 Published online: 20 September 2005 – © Springer-Verlag 2005
Abstract: We investigate a class of area preserving non-uniformly hyperbolic maps of the two torus. First we establish some results on the regularity of the invariant foliations, then we use this knowledge to estimate the rate of mixing. 1. Introduction In recent years the physics community has devoted increasing attention to anomalous properties of physical systems (e.g., anomalous transport, anomalous diffusion, anomalous conductivity, etc.). Such properties have proven relevant in many fields such as thermal conductivity, kinetic equations, plasma physics, etc. and they are widely believed to be dynamical in nature. In fact, such phenomena seem to depend on the weak chaotic properties of the underling dynamics, see [24] and references therein for a detailed discussion. The basic idea is that, while uniformly hyperbolic dynamics gives rise to normal transport properties (consider for example the diffusive behavior in a finite horizon Lorentz gas [4]) non-uniform hyperbolicity gives rise to different behavior (e.g. the anomalous diffusion believed to occur in infinite horizon Lorentz gas [3]) due to weaker mixing properties (e.g. polynomial decay of correlations) and regions in which the motion is rather regular and where the systems spend a substantial fraction of time (sticky regions). Unfortunately, the theoretical understanding of dynamical models with polynomial decay of correlations is extremely limited, hence the necessity to rigorously investigate relevant toy models. The only well understood cases are expanding one dimensional maps with a neutral fixed point. Such maps were proposed as a model of intermittent behavior in fluids ([19]) and have been widely studied. It has been proven that such maps enjoy polynomial decay of correlations with the rate depending on the behavior of the fixed point [13, 22, 8, 9, 20, 5]. In addition, when the decay of correlation is sufficiently slow, the observables do not satisfy the Central Limit Theorem or the Invariance Principle but rather, when properly rescaled, some stable law ([25, 6, 17]).
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Some, more partial, results exist for multidimensional expanding maps [18] as well. Yet, the usual physically relevant models are connected to Hamiltonian dynamics and, to our knowledge, no rigorous results are available in such a situation. The simplest case which retains some Hamiltonian flavor is clearly a two dimensional area preserving map. In fact, mixing area preserving maps of the two dimensional torus with a neutral fixed point (the simplest type of sticky set) has been investigated numerically [1] predicting the possibility of a polynomial decay of correlations. In this paper we consider a class of non-uniformly hyperbolic symplectic maps of the two torus where the uniform hyperbolicity breaks down because of a non-hyperbolic fixed point, (2.1). In fact, the linearized dynamics at the fixed point is a shear, (2.3). We prove that the decay of correlations is polynomial and, more precisely, for a large class of observable is at least n−2 . In addition, we show, see Lemma 10.1, that one cannot expect a much stronger result: it is possible to construct a sequence of observables that essentially saturates our bound. Note that the example treated numerically in [1] is a special case of the present setting. In [1] the authors investigate some specific observables and predict a decay of order n−2.5 . This is not incompatible with our result since specific observables can have a faster than “typical” decay. We are unable, at the moment, to determine if the prediction in [1] is correct or not. The latter considerations clearly emphasize the difficulty to investigate such issues and the strong need for more numerical and theoretical work on the subject. The results of the present paper are based on a precise quantitative investigation of the hyperbolic properties of the system. Note that since the system is not uniformly hyperbolic, the standard results for uniformly hyperbolic systems do not apply, thus the need to develop the theory from scratch. The first step is a careful analysis of the angle between the stable and the unstable direction. The angle turns out to degenerate approaching the origin (where the non hyperbolic fixed point is located). Once such a control is achieved it is possible to obtain a bound on the expansion and contraction in the system. Such expansion turns out to be only polynomial, in contrast with the uniformly hyperbolic case where it is exponential. In turn, the bound on the expansion allows to study the regularity of the stable and unstable foliation. It turns out that they are C 1 away from the origin. This first part of the paper is rather technical but establishes once and for all the piece of hyperbolic theory necessary as a foundation for each further attempt of investigating the statistical properties of the system. To the latter end we apply the simplest, and roughest, of the tools available: a random approximation technique that allows estimating the correlations as a zero noise limit. This constitutes the second, much shorter, part of the paper. As the rate of convergence to equilibrium is of order n−2 , see Theorem 2.4, the Central Limit Theorem holds for zero average observable, see Corollary 2.6, so the model does not exhibit anomalous statistical behavior in this respect. Yet, it clearly exhibits an intermittent behavior and it shows the mechanism whereby slow decay of correlations may arise. The present work emphasizes the need to carry out similar studies in cases where the set producing intermittency has a more complex structure than a simple isolated point. The paper is organized as follows: Sect. 2 details the model and makes precise the results. Section 3 studies the local dynamics at the fixed point and, in particular the properties of its stable and unstable manifolds. This can be achieved in many ways; here we find it most efficient to apply a variational technique. Section 4 establishes a precise bound for the angle between the stable and the unstable direction at each point. As anticipated, such a bound yields an a priori bound on the expansion and contractions rates in the systems, these are obtained in Sect. 5. The latter result suffices to apply
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standard distortion estimates that, in turn, allow to prove precise results on the regularity of the invariant foliation and the holonomies, see Sects. 6 and 7 respectively. Next, in Sect. 8 we introduce a random perturbation of the above map and investigate its statistical properties that, thanks to the added randomness, can be addressed fairly easily. The relevance of the above random perturbation is that the limit of zero noise allows to easily obtain a bound on the rate of mixing in the original map; we do this in Sect. 9. Finally, in Sect. 10, we show that the obtained bound is close to being optimal. The paper ends with Remark 10.2 pointing to the unsatisfactory nature of some of the present results and the need to investigate the related open problems. 2. The Model and the Results For each h ∈ C ∞ (T1 , T1 ) we define the map T : T2 → T2 by1 x + h(x) + y mod 1 T (x, y) = . h(x) + y mod 1
(2.1)
We moreover require the following properties: (1) h(0) = 0 (zero is a fixed point); (2) h (0) = 0 (zero is a neutral fixed point); (3) h (x) > 0 for each x = 0 (hyperbolicity). Note that conditions (2–3) imply that zero is a minimum for h , which forces h (0) = 0;
h (0) ≥ 0.
We will restrict to the generic case (4) h (0) > 0. In order to simplify the discussion we will also assume the following symmetry: (5) h(−x) = −h(x). This means that we can write h(x) = bx 3 + O(x 5 ).
(2.2)
Remark 2.1. Note that two facts implied by the above assumptions are not necessary and could be done away with at the price of more extra work: the hypothesis that there is only one neutral fixed point (finitely many neutral periodic orbits would make little difference) and the symmetry (5). We assume such facts only to simplify the presentation of the arguments. 1 Note that the following formula is equivalent, by the symplectic change of variable q = x − y, p = y, to the map
q +p T(q, p) = p + h(q + p)
mod 1 mod 1
which belongs to the standard map family. Yet, the functions h considered here differ substantially from the sine function which would correspond to the classical Chirikov-Taylor well known example.
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Since the derivative of the map is given by 1 + h (x) 1 DT = , h (x) 1
(2.3)
det(DT ) = 1, thus the Lebesgue measure m is an invariant measure (the maps are symplectic). From now on we will consider the dynamical systems (T2 , T , m). Formula (2.3) and property (3) imply that the cone C+ = {v ∈ R2 | Q(v) := v1 , v2 ≥ 0} is invariant for DT . In addition, it is easy to check that Dξ T 2 C+ ⊂ int C+ ∪ {0} for all ξ ∈ T2 \{0}. From this and the general theory, see [14], follows immediately: Theorem 2.2. The above described dynamical systems are non-uniformly hyperbolic and mixing. Example 1. An interesting concrete example for the above setting is given by the function h(x) := x − sin x. The question remains about the rate of mixing, this is the present topic. Remark 2.3. In the following by C we designate a generic constant depending only on T . Accordingly, its value may vary from one occurrence to the next. In the instances when we will need a constant of the above type but with a fixed value we will use sub-superscripts. Theorem 2.4. For each f, g ∈ C 1 (T2 , R), f = 0, holds true2 f g ◦ T n ≤ C f C 1 g C 1 n−2 (ln n)4 . Remark 2.5. As in other similar cases [13, 10, 16] the logarithmic correction is almost certainly an artifact of the technique of the proof. It could probably be removed by using a more sophisticated (and thus more technically involved) approach. See also Sect. 10. From Theorem 2.4 many facts follow: just to give an example let us mention the following result that can be obtained from Theorem 1.2 in [12]. Corollary 2.6 (CLT). Given f ∈ C 1 , f = 0, the random variable 1 f ◦Ti √ n n−1 i=0
converges in distribution to a Gaussian variable with zero mean and finite variance σ . In addition, σ = 0 iff there exists ϕ ∈ L1 such that f = ϕ − ϕ ◦ T .3 The rest of the paper is devoted to the proof of Theorem 2.4 that will find its conclusion in Sect. 9. The basic fact needed in the proof, a fact of independent interest and made quantitatively precise in Lemma 6.3, is the following. Theorem 2.7. The stable and unstable distributions are C 1 in T2 \{0}. 2 3
In fact, a slightly sharper bound holds, see (9.1). In particular this means that the average of f on each periodic orbit must be zero.
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3. The Fixed Point Manifolds As usual we start by studying the local dynamics near the fixed point. The first basic fact is the existence of stable and unstable manifolds. This is rather standard, yet since we need some quantitative information we will construct them explicitly. Instead of constructing them via the usual fixed point arguments it turns out to be faster to use a variational method. 3.1. A variational argument. Let us consider, in a neighborhood of zero, the function x 1 L(x, x1 ) := (x − x1 )2 + G(x) ; G(x) := h(z)dz. (3.1) 2 0 By setting ∂L = x1 − x − h(x), ∂x ∂L y1 := = x1 − x, ∂x1 y := −
we have (x1 , y1 ) = T (x, y), that is L is a generating function for the map (2.1). Then, for each a ∈ R, we define the Lagrangian La : 2 (N) → R by La (x) :=
∞
L(xn , xn+1 ) + L(a, x1 ).
(3.2)
n=1
The justification of the above definition rests in the following lemma. Lemma 3.1. For each a ∈ (−1, 1), La ∈ C 1 (2 (N)) holds true. In addition, if x ∈ 2 (N) is such that Dx La = 0, then setting x0 = a and yn := xn+1 − xn − h(xn ), we have T n (x0 , y0 ) = (xn , yn ). Proof. First of all (2.2) implies that there exists C > 0 such that |G(x)| ≤ C x 4 . It is then easy to see that La is well defined for each sequence in 2 (N). Next, for each n ∈ N let us define (∇La )n := ∂xn La . Clearly, (∇La )1 (x) = 2x1 − x2 + h(x1 ) − a, (∇La )n (x) = 2xn − xn+1 − xn−1 + h(xn ). Of course, for x ∈ 2 (N), ∇La (x) ∈ 2 (N), it is then trivial to check that Dx La (v) = ∇La (x), v. The last statement follows by a direct computation. By the above lemma it is clear that one can obtain the stable manifolds of the fixed point from the critical points of La ; it remains to prove that such critical points do exist. We will start by considering the case a ≥ 0. Define QB := {x ∈ 2 (N) | |xn −
where A :=
2 b;
c :=
A a.
3 A | ≤ B(n + c)− 2 }, n+c
(3.3)
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It is immediate to check that QB is compact and convex. In addition, if a is sufficiently small, then G is strictly convex on [−2a, 2a] which implies that La |QB is strictly convex. Accordingly, La has minimum in QB , moreover the strict convexity implies that such a minimum is unique, for a fixed. Let us call x(a) the point in QB where La attains its minimum. Lemma 3.2. For a small enough, Dx(a) La = 0. Proof. Suppose that ∂xn La (x(a)) = 0 for some n ∈ N, for example suppose it is neg3 ative. Then x(a) is on the border of QB , say x(a)n = A(n + c)−1 + B(n + c)− 2 ; otherwise we could increase x(a)n and decrease La still remaining in QB , contrary to the assumption. But then ∂xn La (x(a)) = 2x(a)n − x(a)n−1 − x(a)n+1 + h(x(a)n ) 2A A 2B B = − x(a)n−1 − x(a)n+1 + h( ) + + n + c (n + c) 23 n + c (n + c) 23 3bA2 B 2A 2A + + + O((n + c)−4 ) ≥− 7 2 3 [(n + c) − 1](n + c) (n + c) (n + c) 2 3
+ =
3
3
B(n + c)3 {2(1 − (n + c)−2 ) 2 −(1−(n+c)−1 ) 2 −(1 + (n + c)−1 ) 2
(6 −
3
3
(n + c) 2 [(n + c)2 − 1] 2 15 4 )B 7
(n + c) 2
+ O((n + c)−4 ) ≥ 0,
provided a is sufficiently small. We have thus a contradiction. The other possibilities are analyzed similarly. To conclude we need some information on the regularity of x(a) as a function of a. Unfortunately, the implicit function theorem does not apply since D 2 La does not have a spectral gap, yet for our purposes a simple estimate suffices. Lemma 3.3. x1 (a) is a Lipschitz function of a. Moreover, when derivable, |y0 (a) | ≤ C |x(a)0 |. Proof. By Lemma 3.2 it follows, for each a, a sufficiently small ∂xn La (x(a )) = ∂xn La (x(a)) = 0, that is ∂xn La (x(a )) − ∂xn La (x(a)) = ∂xn La (x(a)) − ∂xn La (x(a)) which yields (2 + h (ξ1 ))ζ1 − ζ2 = a − a, (2 + h (ξn ))ζn − ζn+1 − ζn−1 = 0, where ζn := x(a )n − x(a)n and ξn ∈ [x(a)n , x(a )n ]. Notice that, if |ζn | ≥ |ζn−1 |, then |ζn+1 | = |(2 + h (ξn ))ζn − ζn−1 | ≥ 2|ζn | − |ζn−1 | ≥ |ζn |.
(3.4)
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Thus, by induction, if |ζn | ≥ |ζn−1 |, then |ζm | ≥ |ζn−1 | for each m ≥ n, which would imply ζn−1 = 0 since ζ ∈ 2 (N). But then (2 + h (ξn ))ζn = ζn+1 , that is |ζn+1 | ≥ |ζn |. Accordingly, again by induction, ζm = 0 for each m ≥ n − 1. This means that we can restrict ourselves to the case ζn = 0, |ζn | ≥ |ζn+1 |. Hence, |a − a| = |(2 + h (ξ1 ))ζ1 − ζ2 | ≥ 2|ζ1 | − |ζ2 | ≥ |ζ1 |. That is |x(a )n − x(a)n | ≤ |x1 (a ) − x1 (a)| ≤ |a − a|. Finally, summing (3.4) over n ∈ N, ∞
h (ξn )ζn = −ζ1 + a − a.
n=1
Thus, where all the x(a)n are differentiable (a full measure set), |x(a)n | ≤ |x(a)1 | ≤ 1 and x(a)1 = 1 −
∞
h (x(a)n )x(a)n ,
n=1
y(a)0 = −
∞
h (xn )x(a)n .
(3.5)
n=0
Accordingly, |y0 (a) | ≤ 6b
∞
xn2 ≤ C a.
n=0
Clearly, the above lemma implies that, calling (x, γs (x)) the graph of the stable manifold, γs ∈ Lip(−1, 1). The case a ≤ 0 and the unstable manifolds can be treated similarly, yet there exists a faster–and more instructive–way.
3.2. Reversibility. Notice that the map T is reversible with respect to the transformations4 (x, y) := (x, −y − h(x));
1 (x, y) := (−x, y + h(x)).
(3.6)
Indeed, 2 = 21 = Id and T = 1 T 1 = T −1 . Remark 3.4. The reversibility implies that, for x ≥ 0, (x, γu (x)) = (x, γs (x)), and, for x ≤ 0, (x, γu (x)) = 1 (−x, γs (−x)) is the unstable manifold of zero. 4 While the reversibility for is a general fact, the one for depends on the simplifying symmetry 1 hypothesis (5).
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3.3. A quasi-Hamiltonian. To study the motion near the fixed point it is helpful to find a local “Hamiltonian” function. By Hamiltonian function we mean a function that is locally invariant for the dynamics. Such a function can be computed as a formal power series starting with the relation H ◦ T = H . In fact, we are interested only x in a suitable approximation. A direct computation yields that, by defining G(x) := 0 h(z)dz and 1 2 1 1 1 y − G(x) + h(x)y − h (x)y 2 + h(x)2 , 2 2 12 12 H (T (x, y)) − H (x, y) = O(x 8 + y 4 )
H (x, y) :=
(3.7) (3.8)
holds true.5 This approximate conservation law suffices to obtain rather precise information on the near fixed point dynamics.6 The first application is given by the following information on the stable manifold. Lemma 3.5. For x ≥ 0 sufficiently small γs (x) = −A−1 x 2 + O(x 3 ) holds. Proof. Using the notation of Lemma 3.2, for fixed a we get 3 yn = xn+1 − xn − h(xn ) = O (n + c)− 2 .
(3.9)
Hence H (xn , yn ) = O((n + c)−3 ). Using Eq. (3.8) we have H (x0 , y0 ) = H (xn , yn ) + O
n−1
xi8
+ yi4
,
i=0
that is
|H (x0 , y0 )| ≤ C (n + c)
−3
+
∞
(n + c)
−6
≤ C (n + c)−3 + c−5 .
i=0
Taking the limit for n to infinity in the above expression and remembering the definition of c |H (x0 , y0 )| ≤ C x05 5
In fact, setting (x1 , y1 ) := T (x, y), holds 1 1 1 1 h(x)2 − h(x)y − h(x)2 − h (x)y12 − h (x)y13 + h (x)y12 2 2 6 2 1 1 1 1 h (x)y13 − h (x)h(x)y1 + h(x)2 + h (x)y13 − 2 4 12 6 1 8 4 2 4 + h (x)h(x)y1 + O(x + x y + y ). 6
H (x1 , y1 ) − H (x, y) = yh(x) +
6 The reader should be aware that it is possible to do much better, that is to obtain an exponentially precise conservation law, see [11, 2].
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535
Wu Thin sector Ws Fat sector
Fig. 1. The manifolds of the fixed point
3 follows. Since Eq. (3.9) implies |y0 | = O x02 , from (3.7) we have b y02 − x04 + bx03 y0 = O(x05 ), 2 from which the lemma follows.
According to Lemma 3.5, the local picture of the manifolds is given by Fig. 1.
3.4. Manifold regularity. Since in the previous section we have seen that the manifold is Lipschitz curves, we can define the dynamics restricted to the unstable manifold: fu (x) := x + h(x) + γu (x).
(3.10)
Our next task is to obtain sharper information on the manifold’s regularity.7 Lemma 3.6. The unstable manifold of the fixed point is C 2 , apart from zero. Proof. It is clearly enough to show that γu ∈ C 2 apart from zero. To do so call u(x) = γu (x) (the derivative exists almost everywhere since γu is Lipschitz). The tangent vector to the unstable manifold has the form (1, u). On the other hand 1 1 1 + h (fu−1 (x)) 1 := λu (fu−1 (x)) u(x) u(fu−1 (x)) h (fu−1 (x)) 1 1 =: λu (fu−1 (x)) , F (fu−1 (x), u(fu−1 (x))) where := 1 + h (x) + u(x) = fu (x), 1 . F (x, u) := 1 − 1 + h (x) + u
λu (x)
(3.11)
Accordingly, setting xi := fu−i (x), u(xi ) = F (xi+1 , u(xi+1 )) holds. Next, let vi := 2A−1 xi , then a direct computation yields F (xi , vi ) − vi−1 = O(xi3 ), thus |u(xi ) − vi | ≤ |F (xi+1 , u(xi+1 )) − F (xi+1 , vi+1 )| + C xi3 ≤ |u(xi+1 ) − vi+1 | + C xi3 . By induction, and Eq. (3.3), |u(x) − 2A−1 x| ≤ C
∞
3 i=0 xi
≤ C x 2 follows.
7 Of course, the manifold should be as smooth as h, but this result is not needed in the following while we do need an explicit bound on the curvature.
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On the other hand, given a different point z, u(x) − u(z) = F (x1 , u(x1 )) − F (z1 , u(z1 )) = λu (x1 )−1 λu (z1 )−1 (u(x1 ) − u(z1 )) + h (x1 ) − h (z1 )) . holds. Iterating the above equation yields u(x) − u(z) = λu,n (x)−1 λu,n (z)−1 (u(xn ) − u(zn )) n + λu,k (x)−1 λu,k (z)−1 (h (xk ) − h (zk )),
(3.12)
k=1
where λu,n (x) := nk=1 λu (fu−k (x)). Next, let x = a, then, accordingly to Lemma 3.2, Eq. (3.3) and Remark 3.4, we have xi = A(i + c)−1 + O((i + c)3/2 ). This means that, in a sufficiently small neighborhood of zero, and for z sufficiently close to x, λu,n (x) ≥
n
1 + vk − C xi2
k=1 n
≥e
≥
n k=1
2 −3/2 k=1 k+c −C(k+c)
2 1+ − C(k + c)−3/2 k+c
≥ C(A−1 an + 1)2
holds, provided x is close enough to zero. The same estimate holds for λu,n (z). This implies that u is continuous. Indeed, for each ε > 0 choose n0 (x) ∈ N such that λu,n0 (x) (xn )−1 ≤ ε, then |u(x) − u(z)| ≤
n 0 (x)
|h (xk ) − h (zk )| + ε,
k=1
and we can thus choose z close enough to x such that |u(x) − u(z)| ≤ 2ε. Note that this implies the continuity of the λu,n as well. To conclude, we choose n(z) such that C(A−1 an(z) + 1)−4 ≥ |x − z|1+α , for some α > 0, accordingly u(x) − u(z) =
n(z)
λu,k (x)−1 λu,k (z)−1 (h (xk ) − h (zk )) + O(|x − z|1+α ).
k=1
Since the series is uniformly convergent we have u (x) =
∞
λu,k (x)−3 h (xk ),
(3.13)
k=1
from which the lemma follows.8
Remark 3.7. All the above results for the unstable manifold γu have the obvious counterpart for the stable manifold γs that can be readily obtained via reversibility, see Remark 3.4. 8 Remark that to obtain the result on a larger neighborhood it suffices to iterate the unstable manifold forward.
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4. A Narrower Cone Field Here our goal is to estimate the angle between stable and unstable manifolds. + More precisely, we wish to prove that there exist two √constants K+ , K− ∈ R√ such 2 that the cone field C∗ (ξ ) := {(1, u) ∈ R | K− (|x| + |y|) ≤ u ≤ K+ (|x| + |y|)} contains the unstable direction (by reversibility we can also define the stable cone field C∗− ). Proposition 4.1. For each ξ ∈ T2 , E u (ξ ) ∈ C∗ (ξ ), E s (ξ ) ∈ C∗− (ξ ) holds true. The rest of the section is devoted to the proof of Proposition 4.1. Clearly a problem arises only in a neighborhood of zero. Accordingly the first step is to gain a better understanding of the dynamics near zero. 4.1. Near fixed point dynamics. For each δ > 0 let Qδ := [−δ, δ]2 be a square neighborhood of zero. The manifolds of the fixed point divide Qδ into four sectors: two thin and two fat (see Fig. 1). We will discuss explicitly the dynamics in the two sectors below the unstable manifold (the other two being identical by symmetry). Lemma 4.2. For each (x, y) ∈ T2 , y ≤ γu (x), let (xn , yn ) := T n (x, y), then xn ≤ fun (x) ∀n ≥ 0 holds true. Proof. First note that the trajectory will always remain below the unstable manifold. Hence, by induction, xn+1 = xn + h(xn ) + yn ≤ xn + h(xn ) + γu (xn ) ≤ fun (x) + h(fun (x)) + γu (fun (x)) = fun+1 (x). The above lemma will suffice to control the dynamics in the thin sector, more work is needed for the fat one. In fact, when the trajectories are close to the stable or the unstable manifolds, the above result can still be used (possibly remembering reversibility). On the other hand when the trajectory is close enough to zero its behavior is drastically different from the one on the invariant manifolds. To define more precisely the meaning of “close to zero” let us introduce the parabolic sector PM := {(x, y) ∈ Q1 | |y| ≥ Mx 2 }. We consider a backward trajectory starting from x ≤ 0, y ≤ γu (x), the other possibilities follow by reversibility. Let, as usual, (xn , yn ) := T n (x, y), n ∈ Z. Let m+ be the smallest integer for which (x−n , y−n ) ∈ PM , m the largest integer such that x−m ≤ 0, and m− the largest integer for which (x−n , y−n ) ∈ PM . Define, see (3.7), E := H (x−m , y−m ). In addition, define the function ϒE : [−1, 1] → R− , by9 H (x, ϒE (x)) = E. 9
Computing for y ≤ 0 yields
h(x) h(x)2 1 1 − + 2[G(x) + E − h(x)2 ](1 − h (x)) ϒE (x) = − 2 4 12 6 = − 2(E + G(x))(1 + O(x 2 )) + O(x 3 ).
(4.1)
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C. Liverani, M. Martens
2 , Then, by (3.8), and since |y−n | ≥ C x−n
H (x−n , y−n ) − H (x−n , ϒE (x−n )) = H (x−n , y−n ) − H (x−m , y−m ) ≤ C
m
4 y−k
k=n
≤C
m
3 3 |y−k |(x−k−1 − x−k ) ≤ C |y−n ||x−n |.
k=n
Accordingly, for n ≤ m, |y−n − ϒE (x−n )| ≤ C |y−n |2 |x−n |
(4.2)
holds true. √ Lemma 4.3. In the above described situation, setting M = b, the following holds true: (1) fu−n (x) ≤ x−n ∀ n ≤ m; (2) √ fsk (x−n ) ≥ x−n+k√ ∀ n ≥ m− , k ≤ n − m− ; (3) E ≤ |y−n | ≤ 3 E for all n ∈ {m+ , . . . , m− }; 1 (4) m+ ≤ 2A(Eb−1 )− 4 ; 1 1 (5) 2(12Eb)− 4 ≤ m− − m+ ≤ 4(Eb)− 4 . Proof. The first fact is proven as in Lemma 4.2, the second follows by reversibility. Hence, by the results of Sect. 3, for n ≤ m, it follows 2A|x| . (4.3) |x|n + A Next we want to determine the points xm+ and xm− . The idea is to use (4.2) that determines with good precision the geometry of the trajectories. Let x¯ be defined by ϒE (x) ¯ = −M x¯ 2 . Then 1 4 5 2E x¯ = − + O(E 4 ). M 2 − b2 |x−n | ≤ |fu−n (x)| ≤
2 2 On the other hand, since by definition |y−m+ | ≥ Mx−m and |y−m+ +1 | ≤ Mx−m , + + +1 2 3 |y−m+ − Mx−m+ | ≤ C |x−m+ | holds. Hence, by (4.2) it follows
|x−m+ − x| ¯ ≤C
3 x−m +
min{x, ¯ x−m+ }
≤C
3 x−m +
x−m+ − |x−m+ − x| ¯
.
Solving the above inequality yields
Analogously |x−m−
√ 2 |x−m+ − x| ¯ ≤ C x−m ≤ C x¯ 2 ≤ C E. + √ + x| ¯ ≤ C E. From this (3) and (4) easily follows. Finally,
2|x| ¯ ≥ |xm− − xm+ | = |
m−
√ yn | ≥ C(m+ − m− ) E,
n=m+
which implies (5).
We are now ready to refine our knowledge of the stable and unstable direction. Let us fix ∈ (0, 1/2).
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539
4.2. The cone field–Outside Q . The general idea is to take the positive cone field C+ (which is invariant and contains the unstable direction) and to push it forward in order to obtain a narrower cone field. First of all outside Q√ we have (see (3.11)) 1 ≥ F (x, u) ≥ F (x, 0) ≥ 2b, where we have chosen small enough. Hence the cone field C0 = {(1, u) | 1 ≥ u ≥ 2b} is invariant outside Q√ . It remains to understand what happens in a neighborhood of √ the origin of order . Let us define u(ξ ¯ ) by the equation u = F (ξ, u). An easy computation shows −h (x) + h (x)2 + 4h (x) √ u(ξ ¯ )= = 3b|x| + O(x 2 ). 2 By reversibility we can restrict ourselves to the case x ≥ 0; in this case the only possibility to enter the region Q√ is via the fourth quadrant. Note that F (ξ, u) ≥ u, provided 0 ≤ u ≤ u(ξ ¯ ). This means that if ξ ∈ Q√ but T ξ ∈ Q√ , then the lower bound of the cone Dξ T n C0 does not decrease until √ 3bxi ≤ 2b, where (xi , yi ) := T i ξ . Accordingly, the cone field C0 is invariant also in the fourth quadrant, outside the set Q b . Now consider the cone field C1 (ξ ) := {(1, u) | 2b ≤ u ≤ Lu(ξ ¯ )}, 2
3
1
L = (3b)− 2 , for ξ ∈ Q√ \Q
2
b 3
. Clearly, if |x| ≥
√
, then Lu(ξ ¯ ) ≥ 1. Hence, as
the point enters Q√ , the image of C0 is contained in C1 , moreover we have already seen that the lower bound is invariant provided ξi ∈ Q b . Let us follow the upper edge, if 2
3
u ≤ Lu(ξ ¯ i ), then10 F (ξi , u) = F (ξi , u) − F (ξi , u(ξ ¯ i )) + u(ξ ¯ i) (L − 1)u(ξ ¯ i) ≤ + u(ξ ¯ i) (1 + h (xi ) + Lu(ξ ¯ i ))(1 + h (xi ) + u(ξ ¯ i )) ≤ Lu(ξ ¯ i ) − (L2 − 1)u(ξ ¯ i )2 + O(xi3 ) √ ¯ i )2 + O(xi3 ) ≤ Lu(ξ ¯ i+1 ) + 3b|yi+1 | − (L2 − 1)u(ξ ≤ Lu(ξ ¯ i+1 ), √ provided ξi+1 ∈ PM , with M ≤ 3b(L2 − 1), which is fine provided is chosen small enough. The above discussion can be summarized as follows. Lemma 4.4. There exists > 0: For ξ ∈ Q
2
in C0 . In addition, in the set {ξ = (x, y) ∈
b 3
Q√
the unstable distribution is contained
unstable direction is contained in C1 .
\ (PM ∪ Q
2
b 3
) : xy ≤ 0} the
To conclude we need to study what happens in a neighborhood of the origin of order . It is necessary to distinguish two possibilities: one can enter below the stable manifold, and hence be confined in the fat sector, or one can enter above the stable manifold, thereby being bound to the thin sector. We will start with the easy case: the second. 10
Note that this computation holds for all ξi ∈ Q√ \PM .
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C. Liverani, M. Martens
4.3. The cone field–Thin sector. If x > 0, as soon as the trajectory, at some time n, enters Q b we have that (1, u) ∈ C0 implies u ≥ 2b ≥ γu (xn ). Let us consider in 2
Q
2
in Q
b 3
2
3
the cone field C2 := {(1, u) | Lu(ξ ¯ ) ≥ u(x) ≥ γu (x)}. Note that upon entering b 3
such a cone contains C1 .11 Now
F (x, u) ≥ F (x, γu (x)) = γu (x + h(x) + γu (x)) ≥ γu (x + h(x) + y), where we have used that γu (x) ≥ 0 for x ∈ [0, ], provided has been chosen small enough. Lemma 4.5. In the region Q
2
b 3
\ (PM ∪ {ξ = (x, y) ∈ T2 : y ≥ γ u (x) for x >
0; y ≤ γ u (x) for x < 0}) the unstable direction is contained in the cone field C2 . Note that the above lemma suffices for trajectories in the thin sector. The situation is not so simple in the fat sector since the lower bound would deteriorate to zero. A more detailed analysis is needed. For each ξ = (x, y) ∈ T2 , for which the unstable direction is defined, let (1, u(ξ )) be the vector in the unstable direction. Define then λu,n (ξ, u) and Fn (ξ, u) as in formulae (3.11) and (3.12) and similarly define the stable quantities. That is DT −n ξ T n (1, u) =: λu,n (ξ, u)(1, Fn (ξ, u)), Dξ T −n (1, −v) =: µs,n (ξ, v)(1, −Fn− (ξ, v)).
(4.4)
4.4. The cone field–Fat sector. First of all notice that the trajectory can enter Q
2
b 3
either in PM or outside. Since the cone field C2 for x ≥ 0, ξ ∈ PM contains the unstable vector (Lemma 4.5), we have good control on the unstable vector in both cases until we enter in PM . Upon entering PM , we will obtain a very sharp control on the evolution of the edges of the cone. Let ξ ∈ PM , T ξ ∈ PM , and let + − 1 > 0 be the smallest integer such that ξn ∈ PM . By Eq. (3.11), we have un := Fn (ξn , u) =
n
λu,i (ξn−i , un−i )−1 h (xn−i ) + λu,n (ξn , u)−1 u.
i=1
Then, for each n < + , un ≤
n i=1
C h (xn−i ) + u ≤ M
n yi + u i=1
holds true. By Lemma 4.3-(3),(5), it follows that we have, for u ∈ C2 (ξ ), un ≤ C+ |yn |. Moreover, remembering (3.11) and that u ∈ C2 (ξ ), yields √ un ≥ e−2n C+ |yn | u ≥ C u ≥ C− |yn |. Consequently, if for ξ = (x, y) we define the cone C3 (ξ ) = {C− then the above results can be written as follows. 11
Note that in such a case, the trajectory cannot enter in PM .
√
|y| ≤ u ≤ C+
√
|y|},
Convergence to Equilibrium for Intermittent Symplectic Maps
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Lemma 4.6. In PM the unstable direction is contained in the cone field C3 . Finally we have to follow the trajectory outside PM until it exits from Q
2
b 3
. The
upper bound can be treated as before. Not so for the lower bound. Let ξ = (x, y) be a point in the fat sector, x ≤ 0, x−1 ≥ 0. Then, remembering Subsect. 4.1, let E := H (x, y), u0 = 0 and un+1 := F (xn , un ). Clearly, D(x,y) T n C+ ⊂ {(1, u) ∈ R2 | u ≥ un } ∪ {0}. Lemma 4.7. In the situation described above, for each n ∈ N, F (xn , ϒE (xn )) − ϒE (xn+1 ) = O(|yn |3/2 ) holds true. Proof. Notice that since the trajectory lies below the unstable manifold, |y| ≥ C x 2 . It is then convenient to keep track of the orders of magnitude only in terms of powers of y, F (xn , ϒE (xn )) = ϒE (xn ) − ϒE (xn )2 + h (xn ) + O(|yn |3/2 ). On the other hand, differentiating (4.1), one gets ϒE (x) =
h(x) h (x) h(x)2 − − + O(|ϒE |3/2 ). ϒE (x) 2ϒE (x)2 2
Accordingly, by (4.2), h(xn ) + h (xn )ϒE (xn ) h (xn ) h(xn )2 − − + O(|yn |3/2 ) ϒE (xn ) + h(xn ) 2ϒE (xn )2 2 = ϒE (xn ) − ϒE (xn )2 + h (xn ) + O(|yn |3/2 ),
ϒE (xn+1 ) =
from which the lemma easily follows.
Since F is a contraction in u, we can estimate |un − ϒE (xn )| = |F (xn−1 , un−1 ) − F (xn−1 , ϒE (xn−1 ))| + O(|yn−1 |3/2 ) ≤ |un−1 − ϒE (xn−1 )| + O(|yn−1 |3/2 )
n−1 ≤ |ϒE (x0 )| + O |yk |3/2
= O |y0 | +
k=0 n−1
|yk |(xk − xk+1 ) = O(|yn |).
k=0
We have thus proved that there exists a constant C0 > 0 such that un ≥ ϒE (xn ) − C0 ϒE (xn ).
(4.5)
Hence outside PM the image of the cone will belong to the cone field C3 := {(1, u) ∈ R2 | u(ξ ) ≥ ϒE(ξ ) (x)−C0 ϒE(ξ ) (x)}. Note that, upon exiting PM , ϒE (xn )−C0 ϒE (xn ) √ ≥ C− |yn |, provided is chosen small enough. The proposition follows by choosing small enough and remembering Lemmata 4.4, 4.5 and 4.6.
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5. An a Priori Expansion Bound The results of the previous section allow to obtain the following nice estimate on the expansion in the system. Lemma 5.1. There exists K > 0 such that, for each ξ = (x, y) ∈ T2 \{0}, n ∈ N and (1, u) ∈ C∗ (T −n ξ ) = C∗ (ξ−n ), 2 λu,n (ξ, u) ≥ e−K|x| K −1 |x|n + 1 holds true. Proof. Let us fix δ > 0. On the one hand, if the trajectory lies outside of Qδ , then we have an exponential expansion, on the other hand, if the backward trajectory enjoys |x−n | ≥ |x0 |, then Eq. (3.11) and Proposition 4.1 imply 2 (5.1) λu,n (ξ, u) ≥ (1 + K− |x0 |)n ≥ e−K|x0 | K −1 |x0 |n + 1 . We say that the backward orbit of ξ (up to time n) passes p times thru Qδ if {0 ≤ k ≤ n : ξ−k ∈ Qδ } consists of p intervals. The lemma holds for orbits that pass zero-times thru Qδ . Suppose it holds for orbits that pass p times. Let ξ−n ∈ Qδ and let m < n be the last time ξ−m ∈ Qδ , but it passed already p times in Qδ . Moreover, suppose that the lemma holds in Q2δ . Accordingly, for each n ≥ l such that ξ−n ∈ Qδ , 2 2 λu,n (ξ, u) ≥ e−K|x0 | K −1 |x0 |m + 1 e−2Kδ 2K −1 δ(n − m) + 1 2 2 ≥ e−K|x0 | K −1 |x0 |m + 1 K −1 δ(n − m) + 1 2 ≥ e−K|x0 | K −1 |x0 |n + 1 holds, provided δ has been chosen small enough and since it must be n − m ≥ C δ −1 . Thus to prove the lemma it suffices to prove it for the pieces of trajectories in Qδ . There are two cases: a trajectory enters in the thin sector or in the fat one. Let us consider the thin sector first. Set u−j := Fn−j (ξ−j , u). By the usual distortion estimates it follows: λu,n (ξ, u) = ≥ ≥
n
(1 + h (x−j ) + u−j ) ≥
j =1 n
n
(1 + h (x−j ) + γu (x−j ))
j =1 −j
−j
(1 + h (fu (x)) + γu (fu (x))) =
j =1 n
n
−j
(fu ) (fu (x))
j =1
−j +1 −j |fu (x) − fu (x)| − C |fu−j +1 (x)−fu−j (x)| e −j −j −1 (x)| j =1 |fu (x) − fu
≥ e− C |x0 | Now notice that fu−1 (x) ≤
|x0 |2 . |fu−n (x0 )|2
x 1+C x ,
hence fu−n (x0 ) ≤
|x0 | 1+C n|x0 | .
λu,n (ξ, u) ≥ e− C |x0 | (1 + n C |x0 |)2 .
Thus, (5.2)
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For the fat sector we need only to consider the cases in which x0 ∈ Qδ and x0 ∈ Qδ , x0 ≤ 0, since if x0 > 0 the backward trajectory increases the x coordinate. In such cases we have12 λu,n (ξ, u) =
n
(1 + h (x−j ) + u−j ) ≥
j =1 n
(1 + ϒE (x−j ) − C0 ϒE (x−j ))
j =1
j =1 ϒE (x−j )−2 C0 ϒE (x−j )
≥e ≥e
n
−
(x ) ϒE −j j =1 ϒE (x−j ) (x−j −1 −x−j )−3 C0
n
≥ e− C |x0 | e
−
x−n x0
(z) ϒE ϒE (z) dz
n
= e− C |x0 |
j =1 (x−j −1 −x−j )
ϒE (x0 ) . ϒE (x−n )
Let n∗ ∈ N be the last integer for which |G(x−n )| ≥ E, then for n ≤ n∗ , we have E + G(x0 ) − C |x0 | − C |x0 | G(x−n ) + G(x0 ) ≥e . λu,n (ξ ) ≥ e E + G(x−n ) 2G(x−n ) On the other hand comparing the backward motion with the backward motion on the stable manifold, as we did before with the unstable case,13 1 + (n|x0 | C +1)4 ≥ e− C |x0 | (1 + n|x0 | C)2 . λu,n (ξ, u) ≥ e− C |x0 | 2 Next, let us . . . , m}, where m is the larger integer such that x−m ≤ 0; √consider n ∈ {n∗ , √ we have 2 E ≥ ϒE (x−n ) ≥ 2E, x−n
(z) ϒE x−n∗ ϒE (z) dz
ϒE (x−n∗ ) e ≥ e− C |x0 | ϒE (x−n ) 3 ≥ e− C |x0 | (1 + C |x−n∗ |(n − n∗ ))2 , ≥ e− C |x0 | 2
λu,n−n∗ (ξ, u) ≥ e
− C2 |xn∗ | −
where, in the last line, we used Lemma 4.3-(5). By symmetry it will be enough to wait another time m to have |x−2m | ≥ 21 |x0 |, after which the expansion is assured by the estimate (5.1). Next we need to have similar estimates for the stable contraction. By (4.4), µs (v) := 1 + v = µs,1 (ξ, v), v . = h (x − y) + 1+v
F1− (ξ, v) 12 13
Again, E is chosen to be the energy associated to the point of the orbit closer to the origin. Here we use the inequality
1 + (1 + a)4 a ≥ (1 + )2 . 2 2
(5.3) (5.4)
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It is immediate to check that D(x,y) T −1 (1, −v) = µs (v)(1, −F1− ((x, y), v)) and − −i µs,n (ξ, v0 ) := n−1 i=0 µs (v−i ), where v0 = v and v−i−1 := F1 (T ξ, v−i ). An interesting way to transform information on expansion into information on contraction is to use area preserving. Lemma 5.2. Let ξ ∈ T2 , then for each n ∈ N, u, v ≥ 0 let u−n = u, v0 = v, ξ−n = T −n ξ and u−k+1 = F (ξ−k+1 , u−k ), v−k−1 = F − (ξ−k , v−k ). Then µs,n (ξ, v0 )(v−n + u−n ) = λu,n (ξ, u−n )(u0 + v0 ). Proof. Calling ω the standard symplectic form we have µs,n (ξ, v0 )(v−n + u−n ) = ω(Dξ T −n (1, −v0 ), (1, u−n )) = ω((1, −v0 ), Dξ−n T n (1, u−n )) = λu,n (ξ, u−n )(u0 + v0 ). The following is an immediate corollary of Lemmata 5.2 and 5.1. Corollary 5.3. For each ξ = (x, y) ∈ T2 and n ∈ N, µs,n (ξ, v0 ) ≥ e− C |x0 | (C−1 |x0 |n + 1)2
u0 + v 0 u−n + v−n
∀n ∈ N
holds. All the other expansion estimates can be obtained by reversibility. 6. Distributions–Regularity Let (1, u(ξ )), (1, −v(ξ )) be the unstable and stable directions, respectively. We will then use the short hand λu,n (ξ ) := λu,n (ξ, u(ξ−n )) and µs,n (ξ ) := µs,n (ξ, v(ξn )). Lemma 6.1. The unstable distribution is continuous in T2 . Proof. Notice that for ξ = (x, y), ξn := T n ξ , iterating formula (3.11), in analogy with (3.12), u(x) − u(z) = λu,n (x)−1 λu,n (z)−1 (u(x−n ) − u(z−n )) n + λu,k (x)−1 λu,k (z)−1 (h (x−k ) − h (z−k ))
(6.1)
k=1
holds true. By Lemma 5.1, we can take the limit n → ∞ in the above formula provided x = 0, and obtain a uniformly convergent series from which the continuity follows. If ξ = 0 then x−1 = 0 and (3.11) implies u(ξ ) = λu (ξ−1 )−1 h (x−1 ) + λu (ξ−1 )−1 u(ξ−1 ),
(6.2)
hence the continuity at ξ = 0 follows. We are left with the continuity at the origin, but this is already implied by Proposition 4.1.
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This means that we can extend the invariant unstable distribution (that, up to now, where defined–by Pesin theory–only almost everywhere) to a continuous everywhere defined vector field. The same statement holds for the stable vectors by reversibility. Given a continuous vector field there exist integral curves. Since we do not know yet if the vector fields are Lipschitz, it does not follow automatically that from a given point there exists only one integral curve, yet this follows by standard dynamical arguments. Clearly such integral curves are nothing else than the stable and unstable manifolds that are therefore everywhere defined. In addition, remember that, by general hyperbolic theory, the foliations are absolutely continuous; it follows that the above everywhere defined foliations are continuous. Unfortunately, for the following much sharper regularity information is needed; this is obtained in the rest of the section. Let us call ∂ u , ∂ s the derivative along the unstable and the stable vector fields, respectively. Lemma 6.2. The vector field u is C 1 along the unstable manifolds, apart from the origin, moreover |∂ u u(ξ )| ≤ C ∀ξ = 0. Proof. If ξ is outside of a neighborhood of the origin of size δ, then by Lemma 5.1, (6.1) we have, in analogy with the arguments leading to (3.13), ∞ ∞ u −3 |∂ u(ξ )| = λu,k (x) h (xk ) ≤ C (δn + 1)−6 ≤ C . (6.3) n=0
k=1
Since the series converges uniformly the C 1 property follows. To obtain a uniform bound more work is needed. If |ξ | < δ, formula (6.2) implies |∂ u u(ξ )| ≤ λu (ξ−1 )−3 |h (x−1 )| + λu (ξ−1 )−3 |∂ u u(ξ−1 )| =: (ξ−1 , |∂ u u(ξ−1 )|). A simple computation, remembering Proposition 4.1, shows that (ξ, ) ≤ 7b|x| + provided ≥ all ξ .
7b(1+3K− ) . 3K−
≤ 7b|x| + ≤ , 1 + 3|u(ξ )| 1 + 3K− |x|
Accordingly, for ρ large enough, we have |∂ u u(ξ )| ≤ ρ, for
It remains to investigate the regularity of the unstable distribution along the stable direction. Lemma 6.3. The unstable distributions are C 1 along stable manifolds, apart from the origin. Moreover |∂ s u(ξ )| ≤ C
∀ξ = 0.
Proof. Let us fix some arbitrary neighborhood of the origin. Let x, z ∈ W s be outside such a neighborhood. Let W0s be the piece of stable manifold between such two points. Clearly Wns := T n W0s grows for negative n. Let n(x, z) be the largest integer for which 1
s | ≤ |W s | 4 . Our first result is a distortion bound. |W−n 0
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Sublemma 6.4. For each n ≤ n(x, z) and ξ ∈ W0s , s | s | |W−n |W−n ≤ µ . (ξ ) ≤ C s,n |W0s | |W0s |
C−1 holds.
Proof. If the backward orbit spends at least half of the time outside the neighborhood, s grows exponentially fast, hence n(x, z) ≤ C ln |W s |−1 and n(x,z) |W s | ≤ then W−n 0 −i i=0 s is the closest to the C. If this is not the case, the worst possible situation is when W−m origin and all the trajectory lies in the neighborhood. In such a case, letting m := n(x, z), θ (z)(C−1 |z|m + 1)2 |Wms | = µs,m (z)dz ≥ dz, 2θ(T −m z) W0s W0s where θ (ζ ) = u(ζ ) + v(ζ ) is the separation between the stable and the unstable directions at the point ζ and we have used Lemma 5.3. Now Proposition 4.1 and Lemma 4.3-(1) imply θ (T −m z) ≤ C m−1 outside the parabolic sector, while Lemma 4.3-(3,4,5) show that the same estimates remain in PM as well. Accordingly, s | ≥ C m3 |W0s |. |W−m 1
That is m ≤ C |W0s |− 4 , and n(x,z)
1
s |W−i | ≤ n(x, z)|W0s | 4 ≤ C .
i=0
The above estimate readily implies that, for each ξ, η ∈ Ws0 , e− C |W−i | ≤ s
s µs (ξ−i , v(ξ−i )) ≤ eC |W−i | , µs (η−i , v(η−i ))
where we have used Lemma 6.2 for the stable manifold. Accordingly, e− C
m i=0
s | |W−i
≤
from which the lemma readily follows.
m s µs,n (ξ ) ≤ eC i=0 |W−i | , µs,n (η)
By Lemma 5.2 it follows, letting again m := n(x, z),
λu,m (x)−1 λu,m (z)−1 = λu,m (x)−1 λu,m (z)−2 −1 ≤ C θ (x−m )µs,m (x) λu,m (z)θ (z−m )µs,m (z) . s As before the worst case is clearly when W−m is the closest to the origin. In such a s case, consider that at least one of the two end points of W−n(x,z) must be at a diss −n(x,z) z, hence θ(T −n(x,z) z) ≥ tance C |W−n(x,z) | from the fixed point, let us say T s −1 C |W−n(x,z) |. In addition, θ (x−m ) ≥ C m . Indeed, this follows from Lemma 4.3(3,4,5) if the trajectory ends in PM . If the trajectory lies outside PM then it approaches
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547
the origin slower than the dynamics x −C x 2 , which implies x−m ≥ C m−1 . Furthermore by using the above facts, Lemma 5.1, Sublemma 6.4 and the definition of the stopping time m yields |x − z| 43 3 1 −1 −1 ≤ C |x − z|. λu,m (x) λu,m (z) ≤ C |x − z| 4 m m |x − z| 41 Since we know that u is a uniformly continuous function it follows lim λu,m (x)−1 λu,m (z)−1
z→x
|u(T −m x) − u(T −m z)| = 0. |x − z|
Accordingly, by formula (3.12), u (x) = lim
z→x
= lim
z→x
for some ζn ∈ W0 . But
m
λu,n (x)−1 λu,n (z)−1
n=0 m
h (T −n x) − h (T −n z) x−z
λu,n (x)−1 λu,n (z)−1 h (T −n ζn )µs,n+1 (ζn ),
n=0
|h (T −n ζ
n )|
≤ C θ (T −n z), hence
λu,n (x)−1 λu,n (z)−1 θ (T −n x)µs,n+1 (x) ≤ C λu,n (z)−1 , λu,n (x)−1 λu,n (z)−1 θ (T −n z)µs,n+1 (z) ≤ C λu,n (x)−1 . Remembering Sublemma 6.4, the uniform convergence of the series follows and yields the formula ∂ s u(x) =
∞
λu,n (x)−2 µs,n+1 (x)h (T −n x).
(6.4)
n=0
Given the arbitrariness of the neighborhood of zero, the above formula holds for each x = 0 and, since the series converges uniformly, the C 1 property follows. We can now conclude the lemma. By lemma 5.2 |∂ s u(x)| ≤ C
∞
λu,n (x)−2 µs,n (x)θ (T −n x)
n=0
≤C
∞ n=0
follows.
λu,n (x)−1 θ (x) ≤ C
∞ n=0
|x| ≤C (|x|n + 1)2
Remark 6.5. Notice that the symmetrical statements follow by reversibility. The final result on the regularity of the foliations can be stated as follows. Lemma 6.6. The stable and unstable vector fields are C 1 (T2 \ {0}) and, more precisely, for each ξ ∈ T2 \ {0}, |Du(ξ )| ≤ C θ (ξ )−1 .
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Proof. The C 1 property follows from Lemma 19.1.10 of [7]. Then the size of the derivative can be easily estimated by the size of the partial derivatives in the stable and unstable directions divided by the angle between them. Remark 6.7. In fact, it is likely that with a little more work one can show that the foli3 ations are C 2 −ε , but we do not investigate this possibility since it is not needed in the following. 7. Holonomy There exists C1 > 0 such that, given two close stable manifolds W1s , W2s we can define the unstable holonomy u : W1s → W2s by { u (ξ )} := W u (ξ ) ∩ W2s . Let Dr := {ζ = (z1 , z2 ) ∈ R2 : |z1 | ≤ r; |z2 | ≤ r 2 }. Lemma 7.1. For each W1s , W2s disjoint from Dr and ξ ∈ W1s , |J u (ξ ) − 1| ≤ C r −1 u (ξ ) − ξ holds provided u (ξ ) − ξ ≤ C1 r. Proof. Let γs , γ˜s : [−δ, δ] → R2 be W1s , W2s , respectively, parametrized by arc-length. Also, let : [−δ, δ]2 → R2 , be such that (0, 0) = ξ , (s, 0) = γs (s) and (s, t) be the unstable manifold, parametrized by arc-length, of (s, 0) and, finally, (0, ρ) := u (ξ ). Note that (s, t) can be obtained by integrating the unstable vector field starting from (s, 0), hence Lemma 6.6 and the standard results on the continuity with respect to the initial data imply ∈ C 1 . By the transversality of the stable and unstable manifolds there exist τ, σ : [−δ, δ] → R such that (s, τ (s)) = u (γs (s)) = γ˜ (σ (s)) ∈ W2s . Calling η(s) the unit vector perpendicular to γ˜ (s), by the implicit function theorem, it follows τ (s) = −
η(s), ∂ s (s, τ (s)) η(s), ∂t (s, τ (s))
σ (s) = γ˜s (s), ∂ s (s, τ (s)) −
γ˜s (s), ∂t (s, τ (s)) η(s), ∂ s (s, τ (s)) , η(s), ∂t (s, τ (s))
(7.1)
where, clearly, σ (s) = J u (γs (s)). Calling v u (η), η ∈ T2 , the unit vector in the unstable direction at η and v s (η) the stable one, one has ∂t (s, τ (s)) = v u ((s, τ (s))). On the other hand, setting V (s, t) := ∂ s (s, t) − v s ((s, t)), V (s, t) = 0 holds for t = 0, but for t = 0, in general, it will be t V (s, t) = 0. Yet, it is possible to estimate it by differentiating (s, t) = (s, 0) + 0 v u ((s, t ))dt which yields t ∂ s (s, t) = v s ((s, 0)) + Dv u ((s, t ))∂ s (s, t )dt . 0
Lemmata 6.6 and 6.3 imply that Dv u ≤ C r −1 and Dv u v s = |∂ s v u | ≤ C, hence t −1
V (s, t) ≤ C r
V (s, t ) dt + C t. 0
By Gronwall, it follows, provided t ≤ C ρ and ρ ≤ C1 r, for C1 small enough,
V (s, t) ≤ C t.
(7.2)
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Accordingly, by the second of (7.1) and (7.2), it follows |σ (0) − 1| ≤ C r −1 ρ. 8. Random Perturbations The density of a measure with respect to Lebesgue evolves as Lf := f ◦ T −1 . We will then construct a random perturbation by introducing the convolution operator qε (x − y)f (y)dy, (8.1) Qε f (x) := T2
where we assume (1) (2) (3) (4)
q ε (ξ ) := ε−2 q(ε ¯ −1 ξ ); q¯ ∈ C ∞ (R2 , R+ ); q(ξ ¯ )dξ = 1; R2 q(ξ ¯ ) = 0 for each ξ ≥ 1; q(ξ ¯ ) = 1 for each ξ ≤ 21 .
We define then Lε := Qε Lnε , where nε will be chosen later. Notice that qε (x − y)qε (T −nε y − T nε z)f (z)dzdy := L2ε f (x) = T4
(8.2)
T2
Kε (x, z)f (z)dz. (8.3)
We have thus a kernel operator that can be investigated with rather coarse techniques. It turns out to be convenient to define the associated kernel K¯ ε (x, z) := Kε (x, T −nε z) = qε (x − y)qε (T −nε y − z)m(dy). (8.4) T2
For further use let us define Dr := {z = (z1 , z2 ) ∈ T2 | |z1 | ≤ r; |z2 | ≤ r 2 }, Br (ξ ) := {η ∈ T2 ; ξ − η < r}.
(8.5)
The following is a relevant fact used extensively in the sequel. Lemma 8.1. There exists C3 , R > 0 such that, for each δ < R 2 , if Bδ (ξ ) ⊂ DR/2 , then 1 there exists η ∈ B 3 δ (ξ ) such that T n B δ (η) ∩ DR = ∅ for some n ≤ C3 δ − 2 . 4
4
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Proof. If ξ belongs to the first or third quadrant, then T n ξ is escaping from the origin. In such a case, if B 5 δ (ξ ) belongs to the thin sector we choose η ∈ B δ (ξ ), |y| ≥ 38 δ. 8
Clearly, if (x, y) ∈ B δ (η), C x 2 ≥ |y| ≥ 4
δ 4.
2
On the other hand |yn | ≥ n|x|3 while 1
|xn | ≤ |x| + n C |xn |2 . So, if n ≤ C |x|−1 ≤ C δ − 2 , we have |yn | ≥ C |xn |2 . After that we can compare the dynamics with one of the type x → x + C x 2 , hence after a time at 1 most C δ − 2 the T n B δ (η) will exit DR . If the above does not apply, then one can take 4 a ball of radius δ/4 belonging completely to the fat sector and centered at a point in Bδ (ξ ). Then the results of Subsect. 4.1 easily imply the lemma. If, on the contrary, ξ belong to the second or fourth quadrant, then its trajectory may approach the origin in an arbitrary manner (even asymptotically, if the point belongs to the stable manifold). In such a case we can take a point η ∈ Bδ (ξ ) at, at least, a vertical distance 43 δ from the stable manifold and such that B 3 δ (η) ⊂ Bδ (ξ ). Again from the results of Subsect. 4.1 it 4
1
follows that such a ball will exit DR in a time at most C3 δ − 2 .
1
Lemma 8.2. There exist constants σ, C3 > 0 such that if nε = C3 ε − 2 , K¯ ε (x, z) ≥ σ
∀x, z ∈ T2
holds. Proof. It is trivial to see that K¯ ε (x, z) ≥ ε−4 m(Bε/2 (x) ∩ T nε Bε/2 (z)). Accordingly, by Lemma 8.1 there exist two balls of radius 8ε , B¯ 1 ⊂ Bε/2 (z) and B¯ 2 ⊂
Bε/2 (x) that are outside of Dr , r ≥ 2ε , and whose images will be outside of a neigh1
borhood of the origin or order one in a time less than C ε − 2 , forward and backward in time, respectively. Given two unstable manifolds in B1 at a distance larger than crε, ¯ for some appropriate c, ¯ then no stable manifold will intersect both manifolds inside the ball B1 . We can thus consider C r −1 unstable manifolds such that no stable manifolds intersect two of them in B1 . Around each such manifold we can construct a strip by moving along the stable manifold by C ε. We obtain in this way C r −1 disjoint strips each of area C rε2 , whose union covers a fixed fraction of the area of B1 . After a time 1 less than C ε− 2 such strips will be outside a neighborhood of zero, their length may have considerable increase, if so we will subdivide them into strips of length ε. Since now the stable and unstable manifold are at a fixed angle and by the usual distortion arguments, such strips are essentially rectangular. At this point, by Lemma 5.1, it will suffice to wait 1 a time ε− 2 to insure that each such strip will acquire length at least 21 in the unstable direction. We thus iterate for such a time and, if one strip becomes longer than one, we subdivide it into pieces of length between 21 and one. Finally, fix some box of some fixed size C away from the origin with sides approximately parallel either to the stable or to the unstable directions. By mixing it suffices to wait a fixed time to be sure that a fixed percentage of each one of the above mentioned strips will intersect the box. In addition, it is possible to insure that such strips cut the box from one stable side to the other. We can then write m(Bε/2 (x) ∩ T nε Bε/2 (z)) = m(T −nε /2 Bε/2 (x) ∩ T nε /2 Bε/2 (z)) since the same considerations done above for the unstable manifold can be done, iterating
Convergence to Equilibrium for Intermittent Symplectic Maps
551
backward, for the stable manifold. It follows that a fixed percentage of T −nε /2 Bε/2 (x) and a fixed percentage of T nε /2 Bε/2 (z)) will intersect and hence each other. In fact each one of the above constructed strips in the unstable direction will intersect each one of the strips in the stable direction. By the usual distortion estimates, this implies that the intersection among any two such strips has a measure proportional to the product of the measure of the two strips, hence m(Bε/2 (x) ∩ T nε Bε/2 (z)) ≥ C m(Bε/2 (x))m(Bε/2 (z)), and the lemma.
Lemma 8.3. For each f ∈ L1 ,
f = 0,
Lnε f 1 ≤ (1 − σ )n/2 f 1 holds. 2 2 Proof. Note that Lε 1 = L∗ε 1 = 1 and let M+ ε = {x ∈ T | Lε f ≥ 0}; M+ = {x ∈ 2 T | f ≥ 0}, then, since Lε f = f = 0, 2 2
Lε f 1 = 2 dx Lε f = 2 dx dy Kε (x, y)f (y) M+ M+ T2 ε ε dy f (y) dx[Kε (x, y) − σ ] =2 T2 M+ ε dy f (y) dx[Kε (x, y) − σ ] = 2(1 − σ ) dy f (y) ≤2
M+
T2
M+
= (1 − σ ) f 1 .
Let v u,s = (v1u,s , v2u,s ) be the unit tangent vector fields in the unstable and stable direction, respectively. Clearly |∂ u (Li f )| ≤ |∂ u f |, while |∂ s (g ◦ T i ) ≤ |∂ s g|. Lemma 8.4. For each f, g ∈ C 1 (T2 , R) Qε f g − f g ≤ C ε{ f ∞ + ∂ u f L1 (ν) }{ g ∞ + ∂ s g L1 (ν) } holds, where ν is the measure defined by ν(h) :=
dρ
∂Dρ
h.
Proof. It is convenient to introduce the following change of variables. For each x, y ∈ T2 close enough, let us call [x, y] = Wδu (x) ∩ Wδs (y); note that by Lemma 4.1 such a point is always well defined provided d(x, y) ≤ C δ 2 . We consider then the change of variable : T2 × T2 → R2 × T2 , ξ := x − y, η := [x, y].
(8.6)
Due to the absolute continuity of the holonomies the above change of variable is absolutely continuous. Clearly, ∂xu η = 0 since moving along W u (x) does not change the intersection point with W s (y). On the other hand ∂xs η = J xu v s (η), since moving x
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C. Liverani, M. Martens
along W s (x) moves η on W s (y) by an amount determined exactly by the unstable holonomy u between W s (x) and W s (y). By similar arguments and straightforward computations, −1 ∂x1 η = v2u (x) det v s (x), v u (x) J xu v s (η), −1 J xu v s (η), ∂x2 η = v1u (x) det v u (x), v s (x) −1 J ys v u (η), ∂y1 η = v2s (y) det v u (y), v s (y) −1 J ys v u (η). ∂y2 η = v1s (y) det v s (y), v u (y) In fact, calling θ (x) the sine of the angle between stable and unstable directions at the point x and v⊥ the orthogonal unit vector to v, u (x)|, ∂x η = J xu θ (x)−1 |v s (y)v⊥ s ∂y η = J ys θ (y)−1 |v u (x)v⊥ (y)|
holds. Accordingly,
Id −Id = det ∂x η + ∂y η J := det ∂x η ∂y η
s = J xu J ys θ (x)−1 θ (y)−1 v⊥ (y), v u (x)2 = J xu J ys θ (x)−1 θ (y)−1 det v u (x) v s (y)
2
.
(8.7)
Before starting computing we need to collect some facts. √ Sublemma 8.5. If x ∈ ∂D2ρ , ρ ≥ r ≥ C4 ε, then, for C4 large enough, (i) If x − y ≤ ε, then y ∈ Dρ . (ii) x − η ≤ ρε . (iii) If ζ ∈ W u (x) and ζ − x ≤ ρε , then ζ ∈ Dρ . Proof. The first inequality follows since D2ρ has a vertical size 4ρ 2 . Thus 2ρ 2 − ε ≥ 2ρ 2 − C12 ρ 2 ≥ ρ 2 . Such an estimate and Proposition 4.1 imply that the angle between 4
W u (x) and W u (y) is at least 4K− ρ −1 , thus (ii). Finally, Proposition 4.1 implies that, if 2 ζ = (z1 , z2 ), z2 > 2ρ 2 − ρε 2 ≥ ρ 2 . We can now start computing the integral, dxdyf (x)g(y)qε (x − y) = T4
dx
DCc
4r
T2
dyf (x)g(y)qε (x − y)
+O( f ∞ g ∞ r 3 ) ≥ dηdξf (x)g(y)qε (ξ )J −1 (DCc
4r
×T2 )
+ f ∞ g ∞ O(r 3 ). Next, from formula (8.7) and Sublemma 8.5-(i) follows: ε |J (x, y) − 1| ≤ C 2 . ρ
(8.8)
Convergence to Equilibrium for Intermittent Symplectic Maps
Hence, dxdyf (x)g(y)qε (x − y) = T4
553
dηf (η)g(η) + f ∞ g ∞ O
T2
+O
×T2 ) 4r
(DCc
+O
(DCc
4r
×T2 )
Drc
|1 − J
−1
|
dηdξ [f (x) − f (η)]g(y)qε (ξ ) dηdξf (η)[g(y) − g(η)]qε (ξ )
+ f ∞ g ∞ O(r 3 ). To conclude we must compute the various error terms. For each f ∈ L∞ , 2 R ρ dx f (x) = dρ ds f (s) 3 0 1 + ρ ∂Dρ DR holds. Remembering (8.8) and applying Fubini 1 ε |1 − J −1 | ≤ C f ∞ dρ ρ 2 2 ≤ C f ∞ ε. ρ Drc r η
Next, let γu : [−δ, δ] → T2 be the unstable manifold of η, parametrized by arc-length, η and let s(η, ξ ) be such that γu (s(η, ξ )) = x. Recalling Sublemma 8.5, dηdξ [f (x) − f (η)]g(y)qε (ξ ) (D c ×T2 ) C4 r s(η,ξ ) ≤ g ∞ dηdξ dt |∂ u f (γuη (t))|qε (ξ ) (DCc
4r
≤ C g ∞ ≤ C g ∞
×T2 )
Drc ×T2
0
dη dξ |∂ u f (η )|
1
|∂ u f |
dρ ε 0
ξ qε (ξ ) |θ(η )|
∂Dρ
≤ C g ∞ ∂ u f L1 (ν) ε. Analogously, dηdξf (η)[g(y) − g(η)]qε (ξ ) ≤ C f ∞ ∂ s g L1 (ν) ε. Drc ×T2 We can finally collect all the above estimates and obtain dxdyf (x)g(y)qε (x − y) = dηf (η)g(η) T4
T2
+( f ∞ + ∂ f L1 (ν) )( g ∞ + ∂ s g L1 (ν) )O(r 3 + ε) u
1
from which the lemma follows by choosing r = ε 3 .
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C. Liverani, M. Martens
9. Decay of Correlations Here we put together the results of the previous section to prove Theorem 2.4. Let f, g ∈ C 1 (T2 , R), f = 0, then L
knε
fg =
k−1
L
nε i
nε
(L
− Lε )Lk−i−1 fg ε
+
Lkε f g
i=0
=
k−1
(Lnε − Lε )Lk−i−1 f g ◦ T nε i + O(e− C k f 1 g ∞ ) ε
i=0
=
k−1
(Id − Qε )Lnε Lk−i−1 f g ◦ T nε i + O(e− C k f 1 g ∞ ), ε
i=0
where we have used Lemma 8.3. To conclude, by using Lemma 8.4, we need to estimate j j the L1 norm of ∂ u (Lnε Lε f ). Since |DLε f | ≤ C ε−1 |f |∞ , for j > 0,
∂ (L u
nε
Ljε f ) L1 (ν)
C4
≤
√
ε
dρ 0
∂Dρ √ ε
≤ C f ∞
C4 0
|∂ u (Lnε Ljε f )| +
1 C4
dρ ε−1 ρ + C |f |∞
√
dρ ∂Dρ
ε
1 C4
√
|∂ u (Lnε Ljε f )|
dρ ε
∂Dρ
ε −1 ε ρ2
≤ C f ∞ ln ε−1 , where we have used Lemma 5.1 and Sublemma 8.5.14 Thus √ Ln f g = ( f ∞ + ∂ u f L1 (ν) ) g C 1 O(nε 23 ln ε−1 + e− C n ε ).
(9.1)
Clearly the best choice is ε = C(n−1 ln n)2 which implies the theorem. 10. Lower Bound In this section we prove a lower bound. Lemma 10.1. If there exists a sequence γn such that, for each f, g ∈ C 1 , −n f ◦T g− f g ≤ ( ∂ u f L1 (ν) + f L1 (ν) )|g|C 1 γn T2
T2
T2
holds, then there exists C > 0 such that γn ≥ n−2 C . 14 The above estimate is not sharp. With some extra work one could avoid the ln ε −1 , yet this would not change in any substantial way the result, so we chose to keep the presentation as short as possible.
Convergence to Equilibrium for Intermittent Symplectic Maps
555
Proof. Let g ≥ 0 be a smooth function supported away from zero (let us say that the support of g does not intersect D 1 ). Next, let ξ0 = (x0 , y0 ) = ( 21 , y0 ) ∈ W u (0), 2
ξn = T −n ξ0 . For each point η in a neighborhood of ξn let z1 be the distance, along W u (0), between ξn and W u (0) ∩ W s (η), and z2 the distance, along W s (ξn ), between ξn and W u (ξn ) ∩ W s (η). By construction n (η) := (z1 , z2 ) is a map from a neighborhood of ξn to a neighborhood of the origin with the property that the map transforms the stable and unstable foliation into the standard foliation given by the Cartesian coordinates. Clearly, u −1 n (s, 0) = γ (s) (the unstable manifold of the origin parametrized by arc length and u s such that γ (0) = ξn ), while −1 n (0, s) = γn (s) (the unstable manifold of ξn parametrized by arc length). Finally we define fn := (αn βn ) ◦ n with αn (z1 ) := ς (C5 nz1 ), βn (z2 ) := ς (C5 n−1 z2 ), for C5 small enough, and 1 − |x + 1| |x + 1| ≤ 1 ς(x) := 0 |x + 1| > 1. In other words, fn is a function essentially supported on a neighborhood left of ξn of order n−1 in the unstable direction and the stable direction. Accordingly, the supports of fn ◦ T −k and g are disjoint for all k ≤ C n. Lemma 7.1 implies that the support is essentially a rhombus of size n−1 and angle n−1 . Thus we have |g|C 1 ( ∂ u fn L1 (ν) + fn L1 (ν) )γn ≥ fn g ≥ C n−3 . T2
T2
On the other hand, using again Lemma 7.1,
∂ u fn L1 (ν) ≤ C n (α β) ◦ n L1 (ν) ≤ C n−1 , which yields the lemma.
Remark 10.2. Note that the norms in Lemma 10.1 and in Theorem 2.4 (even in the stronger version given by (9.1)) are different. It is not obvious that, putting the L∞ norm instead of the L1 (ν) one keeps the same rate of mixing. More generally, it is well known that in the uniformly hyperbolic setting the smoothness of the function can have an influence on the mixing rate. An analogous effect may arise in the present setting but it remains to be investigated. A related problem that needs to be addressed is the higher dimensional analogue of the present model where the fixed point has different ways of losing full hyperbolicty. It is clear that the present result is only the starting point and not the end of the story. Acknowledgements One of us (C.L.) would like to thank S.Vaienti and M.Benedicks for helpful discussions. In addition, we acknowledge the support from the ESF Program PRODYN, I.B.M., M.I.U.R. and NSF
References 1. Artuso, R., Prampolini, A.: Correlation decay for an intermittent area-preserving map. Phys. Lett. A 246, 407–411 (1998) 2. Benettin, G., Giorgilli, A.: On the Hamiltonian interpolation of near-to-identity symplectic mappings with application to symplectic integration algorithms. J. Stat. Phys. 74(5/6), 1117–1143 (1994) 3. Bleher, P.M.: Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon. J. Stat. Phys. 66(1–2), 315–373 (1992)
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4. Bunimovich, L.A.; Sinai, Ya.G.: Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys. 78(4), 479–497 (1980/81) 5. Gou¨ezel, S.: Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139, 29–65 (2004) 6. Gou¨ezel, S.: Central limit theorem and stable laws for intermittent maps. Probab. Theory Related Fields 128(1), 82–122 (2004) 7. Katok, A., Hasselblatt, B.: Introduction to the Modern Theorey of Dynamical Systems. Encyclpedia of Mathematical and its Applicatios, 54, Cambridge: Cambridge University Press, 1995 8. Hu, H.: Decay of correlations for piecewise smooth maps with indifferent fixed points. Ergodic Theory Dynam. Syst. 24(2), 495–524 (2004) 9. Hu, H.: Statistical properties of some almost hyperbolic systems. In: Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math. 69, Providence, RI: Am. Math. Soc., 2001, pp. 367–384 10. Liverani, C.: Flows, Random Perturbations and Rate of Mixing. Ergodic Theory Dynam. Syst. 18(6), 1421–1446 (1998) 11. Liverani, C.: Birth of an elliptic island in a chaotic sea. Math. Phys. Electronic J. 10, 1 (2004) 12. Liverani, C.: Central Limit Theorem for Deterministic Systems. In: International Conference on Dynamical Systems, Montevideo 1995, a tribute to Ricardo Ma˜ne, Pitman Research Notes in Mathematics Series, 362, F. Ledrappier, J. Levovicz, S. Newhouse (eds.), London: Pitman, 1997 13. Liverani, C., Saussol, B., Vaienti, S.: A Probabilistic Approach to Intermittency. Ergodic Theory Dynam. Syst. 19, 671–685 (1999) 14. Liverani, C., Wojtkowski M.: Ergodicity in Hamiltonian Systems. In: Dynamics Reported 4, Berlin, Heidelberg: Springer, 1995 15. Maume, V.: Projective metrics and mixing properties on towers. Trans. Am. Math. Soc. 353, 3371– 3389 (2001) 16. Pollicott, M.: Rates of mixing for potentials of summable variation. Trans. Am. Math. Soc. 352, 843–853 (2000) 17. Pollicott, M., Sharp, R.: Invariance principles for interval maps with an indifferent fixed point. Commun. Math. Phys. 229(2), 337–346 (2002) 18. Pollicott, M., Yuri, M.: Statistical properties of maps with indifferent periodic points. Commun. Math. Phys. 217, 503–520 (2001) 19. Pomeau, Y., Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189–197 (1980) 20. Sarig, O.M.: Thermodynamic Formalism for Countable Markov Shifts. Ergodic Theory Dyn. Syst. 19, 1565–1593 (1999) 21. Wang, X.J.: Statistical physics of temporal intermittency. Phys. Rev. A 40, 6647 (1989) 22. Young, L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999) 23. Yuri, M.: Decay of correlations for certain multi-dimensional maps. Nonlinearity 9(6), 1439–1461 (1996) 24. Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371(6), 461–580 (2002) 25. Zweim¨uller, R.: Stable limits for probability preserving maps with indifferent fixed points. Stoch. Dyn. 3(1), 83–99 (2003) Communicated by G. Gallavotti
Commun. Math. Phys. 260, 557–577 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1424-4
Communications in
Mathematical Physics
Asymptotically Simple Solutions of the Vacuum Einstein Equations in Even Dimensions Michael T. Anderson1 , Piotr T. Chru´sciel2 1 2
Dept. of Mathematics, S.U.N.Y. at Stony Brook, Stony Brook, N.Y. 11794-3651, USA. E-mail: [email protected] D´epartement de math´ematiques, Facult´e des Sciences, Parc de Grandmont, 37200 Tours, France. E-mail: [email protected]
Received: 6 December 2004 / Accepted: 15 March 2005 Published online: 31 August 2005 – © Springer-Verlag 2005
Abstract: We show that a set of conformally invariant equations derived from the Fefferman-Graham tensor can be used to construct global solutions of vacuum Einstein equations, in all even dimensions. This gives, in particular, a new, simple proof of Friedrich’s result on the future hyperboloidal stability of Minkowski space-time, and extends its validity to even dimensions. 1. Introduction Consider the class of vacuum solutions to the Einstein equations (M , g) in n + 1 dimensions, which are future asymptotically simple, i.e. conformally compact, in the sense of Penrose, to the future of a complete Cauchy surface (S , γ ). A natural method to try to construct such space-times is to solve a Cauchy problem for the compactified, unphysical space-time (M , g), ¯ and then recover the associated physical space-time via a conformal transformation. However, a direct approach along these lines leads to severe difficulties, since the conformally transformed vacuum Einstein equations form, at best, a degenerate system of hyperbolic evolution equations, for which it is very difficult to prove existence and uniqueness of solutions. Friedrich [20, 23] has developed a method to overcome this difficulty in 3 + 1 dimensions, by introducing a system of “conformal Einstein equations” whose solutions include the vacuum Einstein metrics and which transforms naturally under conformal changes. A variation upon Friedrich’s approach, again in 3 + 1 dimensions, has been presented in [6]. In this paper, we develop a different approach to this issue which, besides its simplicity, has the advantage of working in all even dimensions. The method, carried out for vacuum space-times with > 0 in [2], is based on use of the Fefferman-Graham Partially supported by NSF grant DMS 0305865 (MA), and the Erwin Schr¨ odinger Institute, the Vienna City Council, and the Polish Research Council grant KBN 2 P03B 073 24 (PTC)
558
M.T. Anderson, P.T. Chru´sciel
(ambient obstruction) tensor H, introduced in [16]. The tensor H is a symmetric bilinear form, depending on a metric g and its derivatives up to order n + 1, cf. Sect. 2 for further discussion. It is conformally covariant, (of weight n − 1) and metrics conformal to Einstein metrics satisfy the system H = 0.
(1.1)
When n = 3, i.e. in space-time dimension 4, the Fefferman-Graham tensor is the wellknown Bach tensor. The main result of the paper, Theorem 4.1, is the proof of the well-posedness of the Cauchy problem for Eq. (1.1), for Lorentz metrics. This leads to a new proof of Friedrich’s result on the future “hyperboloidal” stability of Minkowski space-time [21] (see Theorem 6.1), and extends the validity of this result to all even dimensions.1 As a corollary we additionally obtain existence of a large class of non-trivial, vacuum, even dimensional space-times which are asymptotically simple in the sense of Penrose, see Theorem 6.2. We further note that in [2] existence of solutions of the Cauchy problem for (1.1) is obtained by pseudo-differential techniques. Here we show that the Anderson-Fefferman-Graham (AFG) equations (1.1) can be solved using an auxiliary, first order, symmetrisable hyperbolic system of equations. This shows that (1.1) is a well posed evolutionary system, directly amenable to numerical treatment. Thus, in spacetime dimension four we provide an alternative to Friedrich’s conformal equations for the numerical construction of global space-times [18, 19, 28]. Our methods do not apply in odd space-time dimensions, where the situation is rather different in any case, as one generically expects polyhomogeneous expansions with halfinteger powers of 1/r, where r is, say, the luminosity distance, compare [11, 26, 27, 30]. 2. The Anderson-Fefferman-Graham Equations Let, as before, n + 1 denote space-time dimension, with n odd. The Fefferman-Graham tensor H is a conformally covariant tensor, built out of the metric g and its derivatives up to order n + 1, of the form H = (∇ ∗ ∇)
n+1 2 −2
[∇ ∗ ∇(S) + ∇ 2 (trS)] + F n ,
(2.1)
where S = Ricg −
Rg g, 2n
(2.2)
and where F n is a tensor built out of lower order derivatives of the metric (see, e.g., [25], where the notation O is used in place of H). It turns out that F n involves only derivatives of the metric up to order n − 1: this is an easy consequence of Eq. (2.4) in [25], using the fact that odd-power coefficients of the expansion of the metric gx in [25, Eq. (2.3)] vanish. (For n = 3, 5 this can also be verified by inspection of the explicit formulae for F 3 and F 5 given in [25].) 1 Once most of the work on this paper was completed we have been informed of the work by Schimming [32], who has done a local analysis, related to ours, of the Cauchy problem for the Bach equations in dimension four. The application of his work to global issues, for instance concerning the vacuum Einstein equations, seems not to have been addressed. We are grateful to R. Beig for pointing out that reference to us.
Asymptotically Simple Even-Dimensional Space-Times
559
The system of equations H=0
(2.3)
will be called the Anderson-Fefferman-Graham (AFG) equations. It has the following properties [25]: (1) The system (2.3) is conformally invariant: if g is a solution, so is ϕ 2 g, for any positive function ϕ. (2) If g is conformal to an Einstein metric, then (2.3) holds. (3) H is trace-free. (4) H is divergence-free. Recall that H was originally discovered by Fefferman and Graham [16] as an obstruction to the existence of a formal power series expansion for conformally compactifiable Einstein metrics, with conformal boundary equipped with the conformal equivalence class [g] of g. This geometric interpretation is irrelevant from our point of view, as here we are interested in (2.3) as an equation on its own. 3. Reduction to a Symmetrisable Hyperbolic System Let g be a Lorentzian metric, let ∇ be a connection (not necessarily g-compatible), and let 2 denote an operator with principal part g µν ∇µ ∇ν (acting perhaps on tensors). Let u be a tensor field, and let ∇ (k) u denote the tensor formed from the k th order covariant derivatives of u. For k ≥ 0 consider the system of equations 2k+1 u = F (x, u, ∇u, ∇ (2) u, . . . , ∇ (2k+1) u) ,
(3.1)
for some smooth F . Here we allow the coefficients of 2 as well as the connection coefficients to depend smoothly upon x as well as upon the collection of fields (u, ∇u, ∇ (2) u, . . . , ∇ (2k+1) u), in particular the metric g is allowed to depend (smoothly) upon those fields. We will assume that (3.1) is invariant under diffeomorphisms, although this is not necessary for some of our results below, such as local existence and local uniqueness of solutions. We want to show that solutions of (3.1) can be found by solving a first order symmetric hyperbolic system of PDEs. The idea of the proof can be illustrated by the following example. Consider the equation 22 u = 0 .
(3.2)
Introducing ψ (0) = u ,
ψ (1) = 2u ,
it is easily seen that solutions of (3.2) are in one-to-one correspondence with solutions of the systems (0) ψ ψ (1) 2 . (3.3) = 0 ψ (1) It is then standard to write a symmetrisable-hyperbolic first order system so that solutions of (3.3) are in one-to-one correspondence with solutions of the first order system with appropriate initial data (compare the calculations in the proof below). Some work is needed when we want to allow lower order derivatives as in the righthand-side of (3.1):
560
M.T. Anderson, P.T. Chru´sciel
Proposition 3.1. There exists a symmetrisable hyperbolic first order system P = H () ,
(3.4)
where P is a linear first order operator and H does not involve derivatives of , such that every solution of (3.1), with (M , g) time orientable, satisfies (3.4). Proof. Let {ea }a=0,... ,n = {e0 , ei }i=1,... ,n be an orthonormal frame for g, with e0 a globally defined unit timelike vector; (such vector fields always exist on time orientable manifolds). We set (j )
(j )
ϕ (j ) = {ϕa1 ...ai }1≤i≤2(k−j ) , where ϕa1 ...ai = ea1 · · · eai 2j u, ϕ = {ϕ (j ) }0≤j ≤k , ψ = {ψ (j ) }0≤j ≤k , where
(3.5) (3.6)
ψ (j ) = 2j u .
(3.7)
Let us derive a convenient system of equations for ϕ. First, (j )
for 1 ≤ i ≤ 2(k − j ) − 1
(j )
e0 ϕa1 ...ai = ϕ0a1 ...ai = L(ϕ (j ) ) ,
(3.8)
where we use a generic symbol L to denote a linear map which may change from line to (j ) line. This gives evolution equations for those ϕa1 ...ai ’s which have a number of indices strictly smaller than the maximum number allowed. We continue by noting that, again for 1 ≤ i + 2j ≤ 2k − 1, we have on the one hand (j ) 2ϕa1 ...ai
=
(j ) −e0 ϕ0a1 ...ai
+
n
(j )
e ϕa1 ...ai + L(ϕ (j ) ) ,
(3.9)
=1
and on the other (j )
2ϕa1 ...ai = ea1 · · · eai 2ψ (j ) + [2, ea1 · · · eai ]ψ (j ) =
(j +1) ϕa1 ...ai
+ L(ϕ
(j )
(3.10)
).
Combining those two equations we obtain (j )
e0 ϕ0a1 ...ai =
n
e ϕa1 ...ai + L(ϕ (j ) , ϕ (j +1) ) . (j )
(3.11)
=1
Note that the condition i +2j ≤ 2k −1 implies j < k so that (3.11) can also be rewritten as (j )
e0 ϕ0a1 ...ai =
n
(j )
e ϕa1 ...ai + L(ϕ) .
(3.12)
=1
Next, for i + 2j = 2k − 1 and for running from 1 to n we write (j )
(j )
(j )
e0 ϕa1 ...ai = e ϕ0a1 ...ai + [e0 , e ]ϕa1 ...ai =
(j ) e ϕ0a1 ...ai
+ L(ϕ
(j )
).
(3.13)
Asymptotically Simple Even-Dimensional Space-Times
The rewriting of (3.12)-(3.13) in the form (j ) (j ) e0 ϕ0a1 ...ai −e1 ϕ1a1 ...ai (j ) (j ) −e1 ϕ0a1 ...ai +e0 ϕ1a1 ...ai .. . +0 (j ) −en ϕ0a1 ...ai +0
561
(j ) · · · −en ϕna1 ...ai +0 +0 = L(ϕ) .. .. . . (j ) · · · +e0 ϕna1 ...ai
(3.14)
makes explicit the symmetric character of (3.12)-(3.13). It is well known that this system is symmetrisable hyperbolic in the sense of [34, Vol. III] when e0 is a nowhere vanishing (j ) vector field.2 This provides the desired system of evolution equations for those ϕa0 a1 ...ai ’s which have the maximum number of indices. (One could also use (3.14) for any number of indices, but (3.8) is obviously simpler.) If we write (3.14) as P˚ ϕ = 0 , where P˚ is a linear first order operator, then the derivatives ea ϕ satisfy a first order symmetrisable-hyperbolic system of equations P˚ ea ϕ = L(ϕ, ∇ϕ) ,
L(ϕ, ∇ϕ) := [P˚ , ea ]ϕ .
(3.15)
0≤i ≤k−1,
(3.16)
The evolution equations for ψ are simply 2ψ (i) = ψ (i+1) , 2ψ
(k)
= F (x, ψ
(0)
,ϕ
(0)
, ∇ϕ
(0)
),
(3.17)
where in (3.17) we have expressed the derivatives of u appearing in (3.1) in terms of ϕ (0) and ∇ϕ (0) using (3.5). By obvious modifications of the calculation starting at (3.9) and ending at (3.14) one can rewrite the left-hand-side of (3.16)–(3.17) as a first order symmetrisable hyperbolic operator acting on the collection of fields (ψ, ∇ψ) := {(ψ (i) , ∇ψ (i) )}0≤i≤k . Setting = (ψ, ∇ψ, ϕ, ∇ϕ) ,
(3.18)
and letting P be the linear part of the system of equations just described, the proposition follows. The interest of Proposition 3.1 relies in the fact, that it is standard to prove existence and uniqueness of solutions of (3.4) when the initial data for are in H s , s ∈ N, for s > n/2 + 1, provided that (M , g) is globally hyperbolic. If g does not depend on ∇ 2k+1 u, then the threshold can be lowered3 to s > n/2.
2 In fact, (3.14) is symmetric hyperbolic in a coordinate system with e = ∂ and e t = 0. However, t 0 i when g depends upon u and its derivatives it is not useful to use such coordinates, as the construction of the Gauss coordinate system leads to differentiability loss. In any case Gauss coordinates are not well adapted to the proof of existence of solutions when g depends upon u. 3 For s > n/2 + 1 the result follows from [34, Vol. III,Theorem 2.3, p. 375]. However, when the symmetric hyperbolic system has the structure considered here, with g not depending upon ∇ 2k+1 u, the proof in [34] applies for s > n/2.
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M.T. Anderson, P.T. Chru´sciel
Now, not every solution of (3.4) will be a solution of (3.1). Let us show that appropriate initial data for (3.4) will provide the desired solutions. When the space-time metric g is independent of u, let S be a spacelike hypersurface in the space-time (M , g). We choose e0 to be a unit time-like vector field normal to S , so that the ei ’s are tangential at S , and we extend e0 off S in some convenient way, which might vary according to the context. Since (3.1) is an equation of order 2k + 2, the associated Cauchy data consist of a set of tensor fields {f(i) }i=0,... ,2k+1 defined on S which provide initial data for (e0 )i u on S : (e0 )i u|S := e0 · · · e0 u|S = f(i) ,
0 ≤ i ≤ 2k + 1 .
(3.19)
i times (i)
For any i ≥ 0 and ≥ 0 we can use (3.19) to calculate formally ψ (i) |S and ψ0 |S by replacing each occurrence of (e0 )j u by f(j ) , e.g., ψ (0) |S = f(0) , e0 ψ (0) |S = f(1) , ψ (1) |S = (2u)|S =
− e 0 e0 u +
n
ei ei u + α eα u + u
i=1
= −f(2) +
n
ei ei f(0) + 0 f(1) +
n
i=1
S
i ei f(0) + f(0) ,
i=1
arising from the detailed structure of 2, and so on. We for some linear maps (i) (i) will write g(0) for the resulting functions ψ (i) |S and g(1) for the resulting functions e0 ψ (i) |S , so that
α ,
(i)
ψ (i) |S = g(0) ,
(i)
e0 ψ (i) |S = g(1) .
Similarly we can calculate (i)
ϕ01 ...0 |S := ϕ
(i) 0 . . . 0 |S ,
factors (0)
where we replace each occurrence of (e0 )j u by f(j ) , e.g. ϕ01 ...0 |S = f() . We will (i)
write h() for the resulting functions, so that (i)
(i)
ϕ (i) |S = h(0) = g(0) ,
(i)
(i)
(i)
ϕ(1) |S = h(1) = g(1) ,
(i)
(i)
ϕ01 ...0 |S = h() .
When g does depend upon u, then the space-time will be built in the process of solving the equations. In the simplest case of g depending only upon u, the procedure just described should be understood as follows: the initial metric g|S is determined by the initial data f0 . We choose an orthonormal basis {ei }i=0,... ,n for g|S , and interpret e0 as the unit normal to S in the space-time that will arise out of the initial data. Thus, f(1) will be interpreted as the value of the normal derivative of u at S , and so on, and the above considerations remain unchanged when this interpretation is used. Proposition 3.2. Let ∈ C(I, H s (O)), s > n/2 + 2k + 2, s ∈ N, be a solution of (3.4) on a globally hyperbolic region I × O with initial data constructed as described above. Then u := ψ (0) is a solution of (3.1) and (3.19).
Asymptotically Simple Even-Dimensional Space-Times
563
Proof. From (3.16) one has ψ (i) = 2i ψ (0) for 0 ≤ i ≤ k. It remains to show that if (0) ϕa1 ...ai = ea1 · · · eai ψ (0) , then (3.17) will coincide with (3.1). This can be proved by a standard calculation. One sets (j )
(j )
χa1 ...ai = ϕa1 ...ai − ea1 · · · eai ψ (j ) , (j )
and using (3.4) one derives a system of equations which show that χa1 ...ai = 0 for the initial data under consideration. However, the computations involved are avoided by the following argument. Suppose, first, that g, F and ∇ are analytic functions of all their variables. Let us denote by f = {f(j ) }0≤j ≤2k+1 the initial data for (3.1); by an abuse of notation we will write f ∈ H s if f(j ) ∈ H s−j for 0 ≤ j ≤ 2k + 1. We note, first, that by using an exhaustion of I × O by compact subsets thereof it suffices to prove the result when I × O is a conditionally compact subset of the domain of definition of the solution. Let fn be any sequence of analytic initial data which converges in H s (O) to f . Let un be the corresponding solution of (3.4); by stability all un ’s will be defined on I × O for n large enough. Similarly, the stability estimates4 for symmetric hyperbolic systems [29] prove that un is Cauchy in C(I, H s (O)) ∩ C 1 (I, H s−1 (O)). The results in [1] show that un is analytic throughout I × O. Let un be a solution of (3.1) on an open neighborhood Un of S in I × O obtained by the Cauchy-Kowalevska theorem. (Note that Un could in principle shrink as n tends to infinity, but it is nevertheless open and nonempty for each n.) Passing to a subset of Un if necessary we can without loss of generality assume that Un is globally hyperbolic. Now, uniqueness of the solutions of the Cauchy problem for (3.4) shows that un coincides with un on Un . Thus un satisfies (3.1) there and thus, by analyticity, everywhere. This shows that maximal globally hyperbolic solutions of (3.1) with analytic initial data are in one-to-one correspondence with maximal globally hyperbolic solutions of (3.4) with the initial data constructed as above. Then, for H s initial data, Proposition 3.2 follows from continuity of solutions upon initial data for (3.4). Finally, if the fields g, ∇ and F are smooth functions of their arguments, they can be approximated by a sequence of fields g(n), ∇(n) and F (n) which are analytic in their arguments. The estimates for (3.4) just described can similarly be used to show that solutions of the approximate problem converge to solutions of the problem at hand both for Eq. (3.4) and (3.1), which finishes the proof. From what has been said so far we obtain Theorem 3.3. Let s > n/2 + 2k + 2, s ∈ N. For any fields s−i f(i) ∈ Hloc (S ) ,
i = 0, . . . , 2k + 1 ,
there exists a unique solution of (3.1) satisfying (3.19). If the metric g does not depend upon ∇ 2k+1 u, then s > n/2 + 2k + 1 suffices. We note that in local coordinate systems (t, x i ) on an open neighborhood U of O ⊂ S of the form U = I × O, with S ∩ U = {t = 0} and O-compact, the solutions are in i s−i u ∈ ∩2k+1 (O)) . i=0 C (I, H 4 Note that I × ∂O is non-timelike by global hyperbolicity, so that integration by parts gives harmless contributions as far as energy estimates are concerned.
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As usual, the lower bound for the local time of existence of the solution does not depend upon the differentiability class s, so in particular smooth initial data provide smooth solutions. In addition the Cauchy problem for (3.1) is well-posed, in that given a pair 1 , f 2 which are close in H s−i (S ), then the solutions u1 , u2 are also of initial data f(i) loc (i) s−i i close in ∩2k+1 i=0 C (I, Hloc (S )). When g does not depend upon u, there exists a unique maximal globally hyperbolic subset O of M , with S being Cauchy for O, on which the solution exists. This is proved by the usual methods. In the quasi-linear case one also has the existence of a maximal globally hyperbolic development of the Cauchy data, giving a space-time (M , g). This follows from the fact that the domains of dependence for the system constructed above are determined by the light-cones of the metric g, so that a proof along the lines of [5], (compare [7,8]), applies.
4. The Cauchy Problem for the AFG Equations The Cauchy problem for (2.3) has a similar form to that for the Einstein equations. Since the system (2.3) is of order n + 1, the initial data consist of an n-dimensional Riemannian manifold (S , γ ), n = 2k + 1 ≥ 3, with n symmetric two-tensors K (i) prescribed on S . The tensor fields K (i) represent the i th time derivative of the metric g in a Gauss coordinate system around S . Thus, in a neighborhood of S , (or more precisely a neighborhood of a bounded domain in S ), one may write g = −dt 2 + γ (t),
(4.1)
where γ (t) is a curve of metrics on S . Setting e0 = −∇t, one has K (i) =
1 i 1 Le0 g|t=0 = ∂ti γ (t)|t=0 . 2 2
(4.2)
In particular K = K (1) is the extrinsic curvature tensor of S in the final space-time (M , g). The set (γ , K (1) , . . . , K (n) ) is not arbitrary, since the equations H(e0 , ·) = 0,
(4.3)
only involve t-derivatives of g up to order n, and so induce (n + 1) equations on (γ , K (1) , . . . , K (n) ). Because (4.3) is diffeomorphism invariant, this is most easily seen in the coordinates (4.1), where g0α = −δ0α , so that (4.3) only involves t-derivatives of gab , a, b ≥ 1, up to order n. The fact that the constraint equations (4.3) are preserved under the evolution follows in the usual way from the equation δH = 0, where δ denotes the divergence of a tensor. To describe the system of n + 1 constraint equations (4.3) in more detail5 , the GaussCodazzi equations for the embedding S ⊂ (M , g) are:
5
Rγ − |K|2 + H 2 = R + 2 Ric(e0 , e0 ),
(4.4)
δK − dH = Ric(e0 , ·),
(4.5)
An explicit form of (4.3) in space dimension 3 can be found in [32]. The parameterisation of the initial data there is rather different from ours.
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where H = trK. In addition, in a Gauss coordinate system (t, x i ) near S , the Raychaudhuri equation gives ∂t H + |K|2 = − Ric(e0 , e0 ).
(4.6)
We first point out that the curvature scalar R = Rg is determined directly by the initial data; this is in contrast to the situation with the Einstein equations, where R is determined by the evolution equations for the metric. Namely, the left side of (4.4) is determined by the initial data, as is Ric(e0 , e0 ), by (4.6). Thus R is determined by γ and K (i) , for i = 1, 2. To describe the form of the “scalar constraint equation” Hµν nµ nν = 0, n = e0 , note that from (2.2) one has trS = (n − 1)Rg /2n. Together with (4.4) and (4.6), this leads to 1−n 2S00 − ∇0 ∇0 trS = (2 + ∇0 ∇0 ) ∂t H n 1 + (2 + (1 − n)∇0 ∇0 ) Rγ + ... , (4.7) 2n where “...” stands for terms which contain less derivatives of the space-time metric. The t-derivatives of the metric of order 4 cancel out, as expected, and one easily finds that the equation Hµν nµ nν = 0 takes the form
n−1 2
Rγ = ρn ,
(4.8)
where, as before, Rγ is the curvature scalar of γ , with = D k Dk the Laplace operator of γ . Finally, ρn is a functional of (γ , K (1) , . . . , K (n) ) which does not involve derivatives of initial data of order n + 1, while the left-hand-side does. This shows in particular that (4.3) is a non-trivial restriction on the initial data. It also follows from (4.7) that ρn will contain terms of the form 1 (n−1) (trK (n−1) ) − D k D l Kkl , n
(4.9)
with other occurrences of K (n−1) there, if any, being also linear with at most one spacederivative. Thus, the scalar constraint equation Hµν nµ nν = 0 can be viewed either as a non-linear equation of order n + 1 which puts restrictions on γ in terms of the remaining data, or as a second order linear PDE for the trace of K (n−1) . One can similarly check that the equation H0i = 0 takes the form of a linear first order PDE for K (n) , with principal part (n)
D i (Kij −
1 trK (n) gij ) , n
(4.10)
where D denotes the Levi-Civita connection of γ . A set (γ , K (1) , . . . , K (n) ) will be called an initial data set if the fields (γ , K (1) , . . . , K (n) ) satisfy the constraint equations (4.3). The collection of initial data sets is not empty, as every solution of the general relativistic constraint equations solves (4.3). Equations (4.3) are preserved by the following family of transformations, related to the conformal invariance of (2.3). Suppose that the data set (γ , K (1) , . . . , K (n) ) arises from a space-time (M , g) satisfying (2.3). Then (γ , K (1) , . . . , K (n) ) satisfies (4.3), and
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if is any strictly positive function on M , then the set (γ˜ , K˜ (1) , . . . , K˜ (n) ) obtained on S from the metric 2 g also satisfies (4.3). For example, if we set ω(j ) := e0 · · · e0 ()|S ,
ω := |S ,
(4.11)
j factors
where e0 is, e.g., a geodesic extension of e0 off S , then it holds that K˜ (1) = ωK (1) + ω(1) γ .
γ˜ = ω2 γ ,
(4.12)
Similar but more complicated transformation formulae hold for K˜ (i) , i ≥ 2, see Appendix A. This leads to a family of transformations preserving (4.3) which are parameterized by n + 1 functions ω, ω(j ) , j = 1, . . . , n, on S , which are arbitrary except for the requirement that ω > 0. Recall that under a conformal transformation of the space-time metric we have 4 4 4n g˜ ij = n−1 gij ⇒ R˜ = − n−1 R − 2g . (4.13) (n − 1) It follows from this formula that for any given ω = |S > 0 and ω(1) = e0 ()|S one can choose ω(2) so that solves the linear wave equation 4n 2g = R (n − 1)
(4.14)
when restricted to S . (This equation is globally solvable in globally hyperbolic spacetimes, but this is irrelevant for the current discussion. Note that solutions of (4.14) might sometimes develop zeros; these are essential in the analysis of the vacuum Einstein equations). As remarked above that the curvature scalar R is determined by the order 2 part of the initial data set. Taking further e0 -derivatives shows that the remaining ω(j ) ’s ˜ together with its may be chosen so that the conformally transformed curvature scalar R, normal derivatives up to order n − 2, vanish at S . Now the conformal and diffeomorphism invariance of (2.3) requires a suitable choice of gauge in order to obtain a well-posed system. As in [2], we use constant scalar curvature for the conformal gauge and harmonic coordinates for the diffeomorphism gauge; the treatment of the conformal gauge is somewhat different here than in [2]. Thus, we require first that R=0.
(4.15)
If (4.15) holds, then (2.3) takes the form (∇ ∗ ∇)
n−1 2
Ric = −F n .
(4.16)
In harmonic coordinates {y α } = {(τ, y i )} with respect to the conformal gauge (4.15), one has 1 Ricαβ = − g µν ∂µ ∂ν gαβ + Qαβ (g, ∂g) , 2 where Q is quadratic in g and ∂g. Applying (∇ ∗ ∇)(n−1)/2 to this and commuting (∇ ∗ ∇)(n−1)/2 with the ∂g terms in Q shows that (4.16) has the form n+1
n αβ 2g 2 gαβ = −F ,
n still has the form (3.1) and 2g = g µν ∂µ ∂ν acts on scalars. where F αβ
(4.17)
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Finally, as initial data for the (gauge-dependent) variables g0α , 0 ≤ α ≤ n, we choose g0α = −δ0α , on S . The data gab and ∂τ gab , a, b ≥ 1, are determined by the initial data γ and K (1) . The derivatives ∂τ g0α are then fixed by the requirement that the coordinates {y β } are harmonic when restricted to S , i.e. 1 2y β = ∂α g αβ + g αβ g µν ∂α gµν = 0 on S . 2
(4.18)
The higher derivatives ∂τi gαβ , i ≤ n, on S are then determined by the given initial data (γ , K (1) , . . . , K (n) ) and by setting to zero the τ -derivatives of (4.18) up to order n − 1. These choices lead to the following: Theorem 4.1. Consider any class (S , [γ , K (1) , . . . , K (n) ]) satisfying the constraint equations (4.3), with s−1 s−n s (γ , K (1) , . . . , K (n) ) ∈ Hloc (S ) × Hloc (S ) × · · · × Hloc (S ) ,
s > n/2 + n + 1, s ∈ N, where (S , γ ) is a Riemannian metric and where the K (i) ’s are symmetric two-covariant tensors; the equivalence class is taken with respect to the transformations of the data discussed above. Then there exists a unique maximal globally-hyperbolic conformal space-time (M , [g]) satisfying (2.3), where [·] denotes the conformal class, and an embedding i:S →M , for which γ is the metric induced on S by g, with K (i) given by (4.2). One can always choose local representatives of [g] by imposing (4.15). Moreover, the Cauchy problem s (S ) × H s−1 (S ) × · · · × H s−n (S ). with such initial data is well-posed in Hloc loc loc Remark 4.2. The inequality s > n/2 + n suffices for existence of unique solutions in local coordinate patches in Theorem 4.1. One expects that the use of local foliations with prescribed mean curvature and space-harmonic coordinates as in [4] should allow one to lower the threshold s > n/2 + n + 1 to s > n/2 + n in this result. Remark 4.3. The conformal space-time (M , [g]) is smooth if the initial data are. Similarly, real-analytic initial data lead to real-analytic solutions. Proof. Given any initial data set [(γ , K (1) , . . . , K (n) )] satisfying the constraint equations, by using the functions ω(j ) , j = 2, . . . , n from (4.11), one can adjust the tensor fields K (i) , i = 2, . . . , n, so that Rg , together with its transverse derivatives up to order n − 2, vanish on S (see the discussion following (4.14)). Note that this holds for any γ ∈ [γ ], so that ω > 0 and ω(1) are otherwise freely specifiable. Solving (4.17) with this initial data, as described in Sect. 3, one obtains a collection of space-time coordinate patches with a solution of (4.17) there when s > n/2 + n. We recall again that the s spaces above. Cauchy problem for (4.17) is well-posed in the Hloc The argument that these gauge choices are preserved, so that (4.15) holds and the coordinates y α remain harmonic in these local space-time coordinate patches generated by (4.17), is rather similar to the one for the Einstein equations [17]; we give details because of some differences in the analytical tools used.
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First recall that (4.17) can be written in the form n−1
n 2g 2 (Rµν − ∇µ λν − ∇ν λµ ) = −Fµν ,
(4.19)
where 1 λµ := − 2g y µ . 2 As g solves (4.19), its obstruction tensor equals n−1 n−1 1 n − 1 n−3 Rg gµν ) − 2g 2 ∇µ ∇ν Rg . (4.20) Hµν = (−1) 2 2g 2 (∇µ λν + ∇ν λµ − 2n 2n The divergence identity ∇µ Hµν = 0 gives then an equation involving λ and Rg : n−1 1 µ 2 2g 2λν + ∇ν (∇ λµ − Rg ) = lower order commutator terms . (4.21) 2 Since H is trace-free, we further have from (4.20) n−1 1 2g 2 (∇ µ λµ − Rg ) = 0 . 2
(4.22)
Because Rg vanishes to order n − 2 at S , and λj vanishes to order n − 1 at S by the discussion following (4.18), the initial data for this equation vanish. It follows, for instance from the work in Sect. 3, that Rg = 2∇ µ λµ .
(4.23)
This can be used to rewrite (4.21) as n+1
2g 2 λν = lower order commutator terms .
(4.24)
In this last equation all commutator terms in (4.21) that involve gradients of Rg have been replaced by derivatives of λ using (4.23). Since g satisfies the constraint equation (4.3) at S , (4.23) and (4.20) imply that λj vanishes to order n at S , so that (4.24) has vanishing initial data. This system has the form considered in Sect. 3, so we conclude that λ vanishes. Hence, the coordinates y α remain harmonic, Rg = 0 by (4.23), and so H = 0 as well, as desired. The usual procedure, as used in the context of the Cauchy problem for Einstein’s equations, then allows one to patch the solutions together provided s > n/2 + n + 1. An argument as in [5] leads to a unique, (up to diffeomorphism) maximal globally hyperbolic manifold (M , [g]), with [g] satisfying (4.16), with an embedding i : S → M , with the desired initial data on i(S ). The well-posedness statement follows immediately from the discussion following Theorem 3.3. Remark 4.4. In general, there will not be a global smooth gauge for (M , [g]) in which Rg = 0. The local coordinate patches where Rg = 0 need not patch together smoothly, preserving Rg = 0, since the initial data ω > 0, ω(1) on local space-like slices S are freely chosen, and so not uniquely determined.
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To see this in more detail, consider for example the de Sitter space-time M = R×S n , with metric gdS = −dt 2 + cosh2 t gS n (1) . This is a geodesically complete solution of the Einstein equations with Rg = n(n + 1), and so satisfies (2.3). The linear wave equation (4.14) with initial data = c1 , ∂t = c2 on S = {t = 0} has a global solution on M given by n−1 1 =− sinh t + d2 n t dt + d1 , 4 cosh 0 for suitable d1 , d2 . One sees that there are no values of c1 , c2 for which > 0 everywhere on M , so that there is no natural global R = 0 gauge for (M , [gdS ]) with such initial data. Consider for example the solution = − n−1 4 sinh t, giving an R = 0 gauge in the region t < 0, which does not extend to {t = 0}. For t = −ε small, the induced metric on S−ε = {t = −ε} is the round metric γδ on S n of small radius δ = δ(ε). To obtain an R = 0 gauge starting at S−ε which extends up to and beyond {t = 0}, one must choose ω to be a large constant. This causes a discontinity in the choice of gauge for the metric, but not in the structure of the conformal class. Remark 4.5. One may also construct the maximal solution (M , [g]) by means of local gauges satisfying Rg = c0
(4.25)
in place of the scalar-flat gauge (4.15), for any c0 ∈ R. The proof of this is the same as before, noting from the form of (4.13) that given ω and ω(1) on S , one can find ω(2) on S such that R˜ = c0 on S . For example, the de Sitter space-time is a geodesically complete solution in the gauge Rg = n(n + 1). However, it is well-known that (M , gdS ) conformally compactifies to the bounded domain in the Einstein static cylinder gE = −dT 2 + gS n (1) where T ∈ (− π2 , π2 ). The metric gE is of course also a solution of (2.3) with RgE = n(n − 1), which is thus a globally hyperbolic extension of (M , gdS ), since T ∈ (−∞, ∞); it is easily seen that this is the maximal solution. Thus, the choice RgE = n(n − 1) gives a global gauge for [gdS ]. The de Sitter metric itself, with gauge Rg = n(n + 1), has conformal factor relative to gE blowing up to ∞ as T → ± π2 . To obtain an extension of [gdS ] past the range (− π2 , π2 ) of gdS requires a rescaling of the large factor to a factor of unit size. 5. AFG Equations vs. Einstein Equations Consider an initial data set (S , γ , K) for the vacuum Einstein equations in n + 1 dimensions, n odd. Thus (S , γ ) is a Riemannian manifold, and the pair (γ , K) satisfies the vacuum constraint equations with cosmological constant ∈ R. Using Einstein’s equations one can formally calculate the derivatives 1 i ∂ γkl |t=0 , 1≤i≤n (5.1) 2 t in a hypothetical Gauss coordinate system near S in which the space-time metric g takes the form −dt 2 + γ (t), as in (4.1). This gives (i)
Kkl :=
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M.T. Anderson, P.T. Chru´sciel
Proposition 5.1. The initial data set for (2.3) so obtained solves the constraints (4.3), and any such globally hyperbolic solution of (2.3) given by Theorem 4.1 is conformally Einstein. Proof. Let g be the associated maximal globally hyperbolic solution of the vacuum Einstein equations. Then g also solves (2.3), and the result follows from the uniqueness part of Theorem 4.1. A space-time with boundary (M , g) ¯ is said to be a conformal completion at infinity of a space-time (M , g) if the usual definition of Penrose is satisfied; that is, there exists a diffeomorphism from M to the interior of M and a function : M → R+ ∪ {0}, which is a defining function for the boundary I := ∂M , such that ¯ . g = ∗ (−2 g) ¯ is H s smooth if g¯ ∈ H s (M ), (in suitable local coorA conformal completion (M , g) dinates for M ). ¯ ω, ω(1) , . . . , ω(n) ) is said to be a smooth conformal completion at A set (S , γ¯ , K, infinity of a general relativistic initial data set (S , γ , K) if (S , γ¯ ) is a Riemannian manifold with boundary, with S = S ∪ ∂S , and with ω, ω(j ) : S → R being smooth-up-to-boundary functions such that ω is a defining function for the boundary S˙ := ∂S , with γ¯ = ω2 γ
(5.2)
on S . Finally K¯ is a smooth-up-to-boundary symmetric tensor field on S such that the equation ω(1) K¯ = ωK + 2 γ¯ ω
(5.3)
holds on S (compare (4.12)). We further assume that the functions ω(i) , 2 ≤ i ≤ n, are such that the fields K¯ (i) , calculated using the K (i) ’s as in (5.1), and the functions ω(i) , defined before (4.12), can be extended by continuity to smooth tensor fields on S . A conformal completion will be said to be H s if γ¯ ∈ H s (S ) and K¯ (i) ∈ H s−i (S ), i = 1, . . . , n. The above conditions are clearly necessary for the existence of a smooth conformal completion a` la Penrose of the maximal globally hyperbolic development of (S , γ , K); we will see shortly that they are also sufficient. We emphasise, however, that Eqs. (5.2) and (5.3) alone, together with the requirement of smoothness of γ¯ and K¯ are not sufficient for the existence of such space-time completions. Indeed, it follows from the results
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571
in [3] that, in dimension 3 + 1, the requirement of smoothness up-to-boundary of the ¯ It would be of interest to work K¯ (i) ’s, i ≥ 2, imposes further constraints on γ¯ and K. out the explicit form of those last conditions, analogously to [3], in all dimensions. We have the following, conformal version of Proposition 5.1; it allows one to repeat several constructions of Friedrich (see [19] and references therein) for vacuum spacetimes with vanishing cosmological constant in all even space-time dimensions: Theorem 5.2. Let (S , γ , K) be a general relativistic vacuum initial data set, = 0, which admits an H s conformal completion at infinity, s > n/2+n+1, with s ∈ N and n odd. Then there exists an H s space-time with boundary (M , g), ¯ equal to the conformal completion at infinity of the unique maximal development (M , g) of (S , γ , K), so that g = −2 g¯ on M , and ˙ . I ⊃S Proof. The proof is essentially identical to that of Proposition 5.1. Consider the initial data (γ , K (1) , . . . , K (n) ) as constructed at the beginning of this section. One can conformally transform them to initial data 1 (i) K¯ kl := ∂si γ¯kl |t=0 , 2
1≤i≤n
in a hypothetical Gauss coordinate system near S in which the space-time metric g¯ takes the form g¯ = −ds 2 + γ¯ (s). Here we use a normalisation as in the proof of Theorem 4.1 or Remark 4.5, requiring the vanishing (or constancy) of Rg¯ . Theorem 4.1 provides a solution of this Cauchy problem, while the fact that this solution is conformally Einstein follows from Proposition 5.1. The choice of the conformal factor transforming (M , [g]) ¯ to the vacuum Einstein solution (M , g) of course depends on the choice of gauge for [g]; ¯ we describe here how is determined at least in the natural settings corresponding to (4.25). Suppose that γ¯ = ω2 γ is a geodesic compactification of (S , γ ), so that for x nearS˙ , ω(x) = distγ¯ (S˙ , x). Such a compactification is uniquely determined by the choice of a boundary metric on the boundary S˙ . Now the value of ω(1) , at the zero level set of ω is determined by the initial data, (compare Eq. (3.13) of [3] in dimension 3 + 1; an obvious modification of that equation holds in all dimensions). Changing time-orientation if necessary, one will have ω(1) = −1 at S˙ and we extend ω(1) to a neighborhood of S˙ in S to have the same value. These data determine the compactification of the initial data set (S , γ , K). Hence, the proof of Theorem 4.1 gives a unique local solution g¯ of (2.3) in the conformal class [g], ¯ with Rg¯ = c0 , for any given c0 , satisfying the initial conditions. Let ϕ = (n−1)/2 , and set g = −2 g¯ = ϕ −4/(n−1) g. ¯ Then ϕ, (and so ), is uniquely determined in a chart for M containing a portion of I˙ by the requirement that ϕ solves the 4n linear wave equation n−1 2g¯ ϕ − Rg¯ ϕ = 0, with initial data ϕ = ω, ∂s ϕ = ω(1) = −1 on S near S˙ (where s is the Gaussian coordinate). Given such a solution , let I be the connected component of the set { = 0 , d = 0} intersecting S˙ . Since, by construction, Ricg = 0, and g¯ is smooth up to I , standard formulas for the Ricci curvature under conformal changes show that has the usual structure on I , in that ∇ is null, and ∇ 2 is pure trace, on I .
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Remark 5.3. A version of Theorem 5.2, and its proof, also holds for de Sitter-type vacuum solutions of the Einstein equations, where > 0. In this case, the completion is at future or past space-like infinity I + or I − ; the Cauchy data for (M , γ¯ ) at I + , consist of the two undetermined terms g(0) , g(n) in the formal Fefferman-Graham expansion for vacuum Einstein solutions with > 0. This gives an alternate proof of one of the results of [2]. The case < 0 leads to initial-boundary value problems. While it is clear that a generalization of Friedrich’s analysis of this case [22] should exist, precise statements require further investigation. 6. Applications to Semi-Global and Global Stability of General Relativistic Initial Value Problems in All Even Space-Time Dimensions Let (S , γ0 ) be the Poincar´e metric on the (n + 1)-dimensional ball, with n ≥ 3 and n odd. Setting K0 = γ0 , the set (S , γ0 , K0 ) is an initial data set for the vacuum Einstein equations, denoted standard hyperboloidal initial data. The maximal globally hyperbolic development (M , g0 ) of (S , γ0 , K0 ) is given by g0 = −dτ 2 + τ 2 γ0 ,
(6.1)
for τ ∈ R+ , with S = {τ = 1}. This space-time is the interior of the future light cone about a point in Minkowski space-time (the “Milne universe”). With respect to the standard smooth conformal compactification of Minkowski space-time as a bounded domain in the static Einstein cylinder, g¯ 0 = −dT 2 + dR 2 + sin2 R gS n−1 (1) ,
(6.2)
(1) (n) the data (γ¯0 , K¯ 0 , . . . , K¯ 0 ) are C ∞ smooth up to the boundary.As noted in Remark 4.5, this choice of gauge is global and satisfies Rg0 = n(n − 1). As an example of application of the results of the previous section, one now easily obtains:
Theorem 6.1. Let S be an n-dimensional open ball, n odd, and consider a general relativistic initial data set (S , γ , K) which admits an H s conformal completion at infinity, s > n/2 + n + 1, s ∈ N. Then there exists ε = ε(n) > 0 such that if the associated data (S , [(γ¯ , K¯ (1) , . . . , K¯ (n) )]) are ε-close in H s × · · · × H s−n to the (1) (n) data (S , [(γ¯0 , K¯ 0 , . . . , K¯ 0 )]) corresponding to standard hyperboloidal initial data, then the maximal globally hyperbolic development (M , g) of (S , γ , K) is causally geodesically complete to the future. The H s conformal compactification (M , g) ¯ is H s close to (M , g¯ 0 ), and extends to a larger H s space-time containing a regular future time-like infinity ι+ for (M , g). ¯ Proof. The standard space-time (M , g0 ) has a conformal compactification to a bounded domain D in the static Einstein cylinder (6.2), where D corresponds to the range of parameters T + R ∈ [0, π], T − R ∈ [0, π ], R ≥ 0. Future null infinity I + is given by
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I + = {T + R = π}, T ∈ ( π2 , π), with future time-like infinity ι+ = {T = π, R = 0}. The future development of (S , γ0 , K0 ) corresponds to the domain D+ = D ∩ {T ≥ π2 }. Clearly, the compactification (D, g¯ 0 ) extends smoothly to a neighborhood D˜ of D as a globally hyperbolic solution of (2.3). The Cauchy data for such an extension are an (1) (n) extension of the standard Cauchy data (S , [(γ¯0 , K¯ 0 , . . . , K¯ 0 )]) past the boundary (1) (n) ˙ . Similarly, the initial data (S , [(γ¯ , K¯ , . . . , K¯ )]) for (M , g) S ¯ extend in H s past ˙ a neighbhorhood of the boundary S and generate a maximal globally hyperbolic space˜ , g), time (M ˜ satisfying (2.3). By the Cauchy stability associated with Theorem 5.2, ˜ , g) ˜ g¯ 0 ), and in particular is an H s for ε small, the solution (M ˜ is close in H s to (D, ˜ . This shows that, to the future of S , extension of (M , g), ¯ where M = { > 0} in M (M , g) has an H s conformal completion, which extends in H s to a neighborhood of I and ι+ . This gives the result. Note that in dimension 3 + 1 the mere requirement [(γ¯ , K¯ (1) , . . . , K¯ (n) )] ∈ H s × · · · × H s−n ,
s > n/2 + n + 1 ,
regardless of any smallness condition, forbids solutions which have logarithmic terms with small powers of 1/r in polyhomogeneous expansions. Thus, (similarly to the results of Friedrich), the above theorem applies for non-generic initial data sets only. Using Corvino-Schoen type constructions together with the above stability result, as in [9], one obtains: Theorem 6.2. There exists an infinite dimensional space of complete, asymptotically simple globally hyperbolic solutions of the Einstein vacuum equations in all even dimensions n + 1 = 2(k + 1), n ≥ 3. Thus, such solutions are geodesically complete both to the future and past, and have a smooth conformal completion at infinity. Proof. The Corvino-Schoen gluing technique [12, 13] can be used, as in [9, 10], to construct static, parity-symmetric initial data on Rn , for any n ≥ 3, which are Schwarzschild with m = 0 outside a compact set, and which are as close to the Minkowskian data as desired. The resulting maximal globally hyperbolic space-time then contains smooth hyperboloids, close to standard hyperbolic initial data, as in Theorem 6.1, both in the future and in the past. In even space-time dimensions the result follows as in the proof of Theorem 6.1. We note that all the space-times constructed in the proof of Theorem 6.2 possess a “complete I ”; this should be understood as completeness of generators of I in the zero-shear gauge, compare [24].6 One expects the above construction to generalise to initial data which are stationary, asymptotically flat outside of a spatially compact set (rather than exactly Schwarzschild there). This would require proving that the resulting space-times have smooth conformal compactifications near ι0 , (in space-time dimension four this follows from [14, 15, 33]), and working out “reference families of metrics” needed for the arguments in [10]. Those results are very likely to hold, but need detailed checking; note that one step of [33] requires dimension four, and that the family of asymptotically flat stationary metrics in higher dimensions might be richer than that in dimension 3 + 1 [31]. 6
We are grateful to H. Friedrich for useful discussions concerning this point.
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Appendix A. The Infinitesimal Invariance Group of the Constraint Equations In this appendix we study the Lie algebra associated to the group of conformal transformations preserving the constraint equations (4.3). This allows one to derive identities which shed light on the structure of those equations. Suppose that H = 0 and that the space-time metric g is rescaled by 2 , where, in a Gauss coordinate system (t, x i ), so that S = {t = 0}, we have =1+ε
ψ(x i ) j t , j!
for some j ≥ 0, and ε > 0 small. In the new Gauss coordinates (t¯, x¯ i ) we thus have 2 (−dt 2 + gij dx i dx j ) = −d t¯2 + g¯ k d x¯ k d x¯ , leading to (∂t t¯)2 − g¯ k ∂t x¯ k ∂t x¯ = 2 ,
(A.1)
∂t t¯∂i t¯ = g¯ k ∂i x¯ k ∂t x¯ , gij = −2 g¯ k ∂i x¯ k ∂j x¯ − ∂i t¯∂j t¯ .
(A.2) (A.3)
By definition of Gauss coordinates we have (t¯, x¯ i ) = (O(t), x i + O(t)), and also (t¯, x¯ i ) = (t + O(ε), x i + O(ε)). Inserting this in the equations above, matching powers in Taylor expansions we find D i ψ j +2 ψ + O(ε 2 t 2j +2 ) . t j +1 + O(ε 2 t 2j +3 ) , x¯ i = x i + ε t (j + 1)! (j + 2)! At the right-hand-side of (A.3) we have variations related to the fact that all the quantities there are evaluated at the point x¯ = x + ε × (·) + . . . ; to first order; this produces a Lie derivative-type contribution. Next, there are variations related to the fact that t¯ = t + ε × (·) + . . . . Each term t¯i K¯ (i) / i! in (twice) the Taylor expansion of g¯ ij at t = 0 gives then a contribution to the right-hand-side of (A.3) equal to K¯ (i) ψ i − 2j − 2 K¯ (i) −2 (t + ε t j +1 )i + O(ε 2 ) = t i + εψt i+j + O(ε 2 ) . (j + 1)! i! (j + 1)! i! From this we can calculate the coefficients of an expansion in powers of t of the righthand-side of (A.3); inverting those relations, for j = 0 this leads to (1 − ε(i − 2)ψ)K (i) + O(ε 2 ), i = 2; (A.4) K¯ (i) = K (i) − 4ε LDψ g + O(ε 2 ), i = 2, t¯ = t + ε
where L denotes a Lie derivative; note that LDψ g is twice the Hessian of ψ. For j > 0 we obtain instead K¯ (i) = K (i) , 0 ≤ i < j , K¯ (j ) = K (j ) + εψg , (i − 2j − 2)(j + i)! K¯ (j +i) = K (j +i) − εψK (i) 2(j + 1)!i! O(ε2 ), 1 ≤ i = 2; + − 4ε LDψ g + O(ε 2 ), i = 2.
(A.5) (A.6) (A.7)
The terms proportional to ε in those equations describe the desired infinitesimal action.
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We can view H0µ as functions of the metric, its derivatives, and of the symmetric (i) derivatives D(1 ...s ) Kk , with i = 1, . . . , n. As Hµν is a differential operator in the space-time metric of order n + 1, the possibly non-trivial contributions arise from those (i) D(1 ...s ) Kk for which s + i ≤ n + 1. Differentiating the equations 0 = H0µ with respect to ε, at ε = t = 0 for j > 0 one obtains 0=
n+1−j
∂H0µ
s=0
∂D(1 ...s ) Kk
1 (i − 2j − 2)(j + i)! − 2 (j + 1)!i! n−j
n+1−i−j
−
1 2
∂H0µ (j +2)
s=0
∂D(1 ...s ) Kk
∂H0µ
(i)
(j +i)
s=0
i=1
n−1−j
(A.8)
g D ψ (j ) k 1 ...s
∂D(1 ...s ) Kk
D1 ...s (ψKk )
D1 ...s k ψ .
At any point x the derivatives D1 ...s ϕ(x) = D(1 ...s ) ϕ(x) can be chosen independently, which leads to various identities. The simplest one is obtained for j = n, then the last two lines give a vanishing contribution. We parameterise the K (i) ’s by their trace-free part and by trace, obtaining 0=
∂H0µ , ∂D1 ...s trK (n)
s≥0;
(A.9)
this gives a check of the general form of (4.10). When j = n − 1 the last line in (A.8) is zero again, and we obtain 0=
2 s=0
∂H0µ
g D ψ (n−1) k 1 ...s
∂D(1 ...s ) Kk
∂H0µ 1 (1) + (2n − 1) D (ψKk ) (n) 1 ...s 2 ∂D1 ...s K 1
s=0
=
k
2
∂H0µ D ... ψ ∂D1 ...s trK (n−1) 1 s s=0 ∂H ∂H0µ 1 0µ (1) (1) (1) ψK + (K D ψ + ψD K ) . + (2n − 1) k k k (n) (n) 2 ∂Kk ∂D Kk
The vanishing of the coefficients in front of the second derivatives of ψ leads to the identity 0=
∂H0µ , ∂D1 2 trK (n−1)
(A.10)
consistently with (4.9). References 1. Alinhac, S., M´etivier, G.: Propagation de l’analyticit´e des solutions de syst`emes hyperboliques non-lin´eaires. Invent. Math. 75, 189–204 (1984) 2. Anderson, M.T.: Existence and stability of even dimensional asymptotically de Sitter spaces. http://arxiv.org/list, gr-qc/0408072, 2004 (to appear in Annales Henri Poincar´e)
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3. Andersson, L., Chru´sciel, P.T.: On “hyperboloidal” Cauchy data for vacuum Einstein equations and obstructions to smoothness of Scri. Commun. Math. Phys. 161, 533–568 (1994) 4. Andersson, L., Moncrief, V.: Elliptic-hyperbolic systems and the Einstein equations. Annales Henri Poincar´e 4, 1–34 (2003) 5. Choquet-Bruhat, Y., Geroch, R.: Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys. 14, 329–335 (1969) 6. Choquet-Bruhat, Y., Novello, M.: Syst`eme conforme r´egulier pour les e´ quations d‘Einstein. C. R. Acad. Sci. Paris 205, 155–160 (1987) 7. Choquet-Bruhat, Y., York, J.W.: The Cauchy problem. In: General Relativity and Gravitation – the Einstein Centenary (A. Held, ed.), London-New York: Plenum, 1979, pp. 99–160 8. Chru´sciel, P.T.: On completeness of orbits of Killing vector fields. Class. Quantum Grav. 10, 2091– 2101 (1993) 9. Chru´sciel, P.T., Delay, E.: Existence of non-trivial asymptotically simple vacuum space-times. Class. Quantum Grav. 19 L71–L79 (2002), erratum-ibid, 3389 10. Chru´sciel, P.T., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. M´em. Soc. Math. de France. 94, 1–103 (2003) 11. Chru´sciel, P.T., Lengard, O.: Solutions of wave equations in the radiating regime. Bull. Soc. Math. de France 133, 1–72 (2005) 12. Corvino, J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214, 137–189 (2000) 13. Corvino, J., Schoen, R.: On the asymptotics for the vacuum Einstein constraint equations. http//arxiv.org/list/gr-qc/0301071, 2003 14. Dain, S.: Initial data for stationary space-times near space-like infinity. Class. Quantum Grav. 18, 4329–4338 (2001) 15. Damour, T., Schmidt, B.: Reliability of perturbation theory in general relativity. J. Math. Phys. 31, 2441–2453 (1990) ´ Cartan et les math´ematiques d’au16. Fefferman, C., Graham, C.R.: Conformal invariants. In: Elie ´ Cartan, S´emin. Lyon 1984, Ast´erisque, No.Hors S´er. jourd’hui, The mathematical heritage of Elie 1985, pp. 95–116 17. Four`es-Bruhat, Y.: Th´eor`eme d’existence pour certains syst`emes d’´equations aux d´eriv´ees partielles non lin´eaires. Acta Math. 88, 141–225 (1952) 18. Frauendiener, J.: Some aspects of the numerical treatment of the conformal field equations. In: Proceedings of the T¨ubingen Workshop on the Conformal Structure of Space-times, H. Friedrich, J. Frauendiener, (eds), Springer Lecture Notes in Physics 604, Berlin-Heidelberg-New York: Springer, 2002, pp. 261-282 19. Friedrich, H.: Conformal Einstein evolution. In: Proceedings of the T¨ubingen Workshop on the Conformal Structure of Space-times, H. Friedrich, J. Frauendiener, (eds), Springer Lecture Notes in Physics 604, Berlin-Heidelberg-New York: Springer, 2002, pp. 1–50 20. Friedrich, H.: Cauchy problems for the conformal vacuum field equations in general relativity. Commun. Math. Phys. 91, 445–472 (1983) 21. Friedrich, H.: On the existence of n–geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure. Commun. Math. Phys. 107, 587–609 (1986) 22. Friedrich, H.: Einstein equations and conformal structure: existence of anti-de-Sitter-type spacetimes. J. Geom. Phys. 17, 125–184 (1995) 23. Friedrich, H.: Einstein’s equation and geometric asymptotics. In: Gravitation and Relativity: at the turn of the Millennium, N. Dahdich, J. Narlikar (eds.), Proceedings of GR15, 1997, Pune, India: IUCAA, 1998, pp. 153-176 24. Geroch, R., Horowitz, G.: Asymptotically simple does not imply asymptotically Minkowskian. Phys. Rev. Lett. 40, 203–206 (1978) 25. Graham, C.R., Hirachi, K.: The ambient obstruction tensor and Q-curvature. (2004) http://arvix.org/list/math.DG/0405068v1, 2004 26. Hollands, S., Ishibashi, A.: Asymptotic flatness and Bondi energy in higher dimensional gravity. J. Math. Phys. 46, 022503 (2005) 27. Hollands, S., Wald, R.M.: Conformal null infinity does not exist for radiating solutions in odd spacetime dimensions. Class. Quant. Grav. 21, 5139–5146 (2004) 28. Husa, S.: Numerical relativity with the conformal field equations. In: Proceedings of the T¨ubingen Workshop on the Conformal Structure of Space-times, H. Friedrich, J. Frauendiener, (eds), Springer Lecture Notes in Physics 604, Berlin-Heidelberg-New York: Springer, 2002, pp. 159–192 29. Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Rational Mech. Anal. 58, 181–205 (1975) 30. Lengard, O.: Solutions of the Einstein’s equation, wave maps, and semilinear waves in the radiation regime. Ph.D. thesis, Universit´e de Tours, 2001, http://www/phys.univ-tours.fr/∼piotr/papers/batz, 2001
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31. Myers, R.C., Perry, M.J.: Black holes in higher dimensional space-times. Ann. Phys. 172, 304–347 (1986) 32. Schimming, R.: Cauchy’s problem for Bach’s equations of general relativity. In: Differential geometry (Warsaw, 1979), Banach Center Publ., Vol. 12, Warsaw: PWN, 1984, pp. 225–231 33. Simon, W., Beig, R.: The multipole structure of stationary space–times. Jour. Math. Phys. 24, 1163– 1171 (1983) 34. Taylor, M.E.: Partial differential equations. New York, Berlin, Heidelberg: Springer, 1996 Communicated by G.W. Gibbons
Commun. Math. Phys. 260, 579–612 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1423-5
Communications in
Mathematical Physics
Global Geometric Deformations of Current Algebras as Krichever-Novikov Type Algebras Alice Fialowski1 , Martin Schlichenmaier2 1
Department of Applied Analysis, E¨otv¨os Lor´and University, P´azm´any P´eter s´et´any 1, 1117 Budapest, Hungary. E-mail: [email protected] 2 Universit´e du Luxembourg, Laboratoire de Mathematiques, Campus Limpertsberg, 162 A, Avenue de la Faiencerie, L-1511 Luxembourg. E-mail: [email protected] Received: 6 December 2004 / Accepted: 25 April 2005 Published online: 6 September 2005 – © Springer-Verlag 2005
Abstract: We construct algebraic-geometric families of genus one (i.e. elliptic) current and affine Lie algebras of Krichever-Novikov type. These families deform the classical current, respectively affine Kac-Moody Lie algebras. The construction is induced by the geometric process of degenerating the elliptic curve to singular cubics. If the finitedimensional Lie algebra defining the infinite dimensional current algebra is simple then, even if restricted to local families, the constructed families are non-equivalent to the trivial family. In particular, we show that the current algebra is geometrically not rigid, despite its formal rigidity. This shows that in the infinite dimensional Lie algebra case the relations between geometric deformations, formal deformations and Lie algebra twocohomology are not that close as in the finite-dimensional case. The constructed families are e.g. of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory. The algebras are explicitly given by generators and structure equations and yield new examples of infinite dimensional algebras of current and affine Lie algebra type.
1. Introduction Deforming a given mathematical structure is a tool of fundamental importance in most parts of mathematics, in mathematical physics, and physics. Via deforming the object into a “similar” object, we obtain additional information on this object. Moreover, via deformations we can approach the problem whether we can equip the set of mathematical structures under consideration (maybe up to certain equivalences) with the structure of a topological or even geometric space. If this is the case, the space is called a moduli space for these structures. For a fixed object the deformations of this object should reflect the local structure of the moduli space at the point corresponding to the object. Typically, to a deformation of a fixed object there is assigned a certain cohomology class in a suitable cohomology space which is associated to the object.
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In this article we deal with deformations of algebraic structures, more specifically with deformations of Lie algebras, in particular with such of infinite dimension. One-parameter deformations of arbitrary rings and associative algebras, and the related cohomology questions, were first investigated by Gerstenhaber, in a series of articles [12]. The notion of deformation was applied to Lie algebras by Nijenhuis and Richardson [21, 22] and generalized to formal deformations by Fialowski [5, 6]. The cohomology space related to the deformations of a Lie algebra L is the Lie algebra two-cohomology H2 (L, L) of L with values in the adjoint module. Indeed, as long as this space is finite-dimensional, it gives all infinitesimal deformations (i.e. there exists a universal family of infinitesimal deformations). Even more, in this case there exists a versal family for the formal deformations with base in this cohomology space, see results of Fialowski [5, 6], Fialowski and Fuchs [9], from where Theorem 2.15 below is quoted. In the theory of compact complex manifolds the vanishing of the corresponding cohomology space implies rigidity of the manifold, i.e. the manifold can be deformed only trivially, see [19, 23, Thm.4.4]. This lead to the impression that the vanishing of the relevant cohomology spaces will imply rigidity with respect to deformations also in the case of other structures. In an earlier paper [10] the authors showed that this hope is too naive. See also its introduction for more background information. There we considered the (infinite dimensional) Witt algebra (respectively its universal central extension, the Virasoro algebra) and constructed algebraic-geometric deformations of it (each of them parameterized by the points of the affine line) using Krichever-Novikov vector field algebras. These deformations are non-trivial, only the special element in these families will be isomorphic to the Witt algebra, despite the fact that for the Witt algebra the twocohomology space vanishes [7]. In fact, in the case of infinite dimensional Lie algebras the vanishing of the two-cohomology only implies infinitesimal and formal rigidity. Formal rigidity essentially means that every deformation over the algebra of formal power series in finitely many variables is equivalent to a trivial family (see Sect. 2 for precise definitions). Our example shows that the Witt (or Virasoro) algebra despite its formal rigidity is geometrically not rigid. This situation is peculiar for the case of infinite dimensional Lie algebras. For finitedimensional Lie algebras we have strong relations between the different concepts of rigidity. In particular, if a finite-dimensional Lie algebra is non-singular (see Sect. 2) it is infinitesimally, formally, geometrically, and analytically rigid if and only if the two-cohomology space vanishes. In this article we elaborate further on these phenomena. We will consider the case of current algebras g = g⊗C[z−1 , z] and their central extensions g, the affine Lie algebras. Here g is a finite-dimensional Lie algebra. Given an invariant, symmetric bilinear form β, the central extension g is the vector space g ⊕ t C, endowed with the Lie bracket −n [x ⊗ zn , y ⊗ zm ] = [x, y] ⊗ zn+m − β(x, y) · n · δm · t, [t, g] = 0, x, y ∈ g, n, m ∈ Z.
These algebras are of fundamental importance in a number of fields. They supply examples of infinite dimensional algebras which are still accessible to a structure theory. They appear as gauge algebras in Conformal Field Theory (CFT), [1]. More generally, they are symmetry algebras of infinite-dimensional systems, integrable systems, etc. More examples can be found in the book of Kac [16]. For simple finite-dimensional Lie algebras g the central extensions g, the associated affine Lie algebras, are the Kac-Moody algebras of untwisted affine type.
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For g finite-dimensional and simple (and hence rigid) it was shown by Lecomte and Roger [20] that the current algebra g remains formally rigid 1 . We will exhibit again natural algebraic families of algebras containing g as a special element but all other members will not be isomorphic to g. In particular, these families can not be algebraic-geometrically equivalent to the trivial family. Hence, the current algebra g will be geometrically not rigid. All these families can be extended to families of centrally extended algebras and yield in this way nontrivial deformations of the affine algebra g. The families constructed appear as families of higher-genus multi-point current algebras of Krichever-Novikov type, see Sect. 3 for their definition. Hence, they are not just abstract families, but families obtained by a geometric process. The obtained results are not only of importance in the deformation theory of algebras, but also in the fields in which current and affine algebras play a role. In particular, they are of relevance in two-dimensional CFT and its quantization. In the process of quantization of conformal fields one has to consider families of algebras and representations over the moduli space of compact Riemann surfaces (or equivalently, of smooth projective curves over C) of genus g with N marked points. Models of most importance in CFT are the Wess-Zumino-Witten-Novikov models (WZWN). Tsuchiya, Ueno, and Yamada [37] gave a sheaf version of WZWN models over the moduli space. Based on the global operator fields introduced by Krichever and Novikov [18] and further generalized by Schlichenmaier [26–29], Schlichenmaier and Sheinman [35, 36] developed a global operator version. The algebras used consist of meromorphic objects on a Riemann surface which are holomorphic outside a finite set A of N points. The set A is divided into two disjoint subsets I and O. With respect to some possible interpretation of the Riemann surface as the world-sheet of a string, the points in I are called in-points, the points in O are called out-points, corresponding to incoming and outgoing free strings; the world-sheet itself corresponds to possible interaction. In the context of WZWN models of particular interest is the situation I = {P1 , . . . , PK }, the marked points we want to vary, and O = {P∞ }, a reference point. We obtain families of algebras over the moduli space Mg,N of curves of genus g with N = K + 1 marked points, and we are exactly in the middle of the main subject of this article. Starting from the finite-dimensional symmetry algebra (i.e. the Lie algebra g), families of higher genus affine algebras (see Sect. 3 for their definition) and their representations are constructed, associated to the geometric data corresponding to the point in Mg,N . The bundle of conformal blocks appears as a bundle of coinvariants. Now clearly, the following question is fundamental. What happens if we approach the boundary of the moduli space? The boundary components correspond to curves with singularities. Resolving the singularities yields curves of lower genera. By geometric degeneration we obtain families of (Lie) algebras containing a lower genus algebra (or sometimes a subalgebra of it), corresponding to a suitable collection of marked points, as a special element. Or reverting the perspective, we obtain a typical situation of the deformation of an algebra corresponding in some way to a lower genus situation, containing higher genus algebras as the other elements in the family. Such kind of geometric degenerations are fundamental if one wants to prove a Verlinde type formula via a factorization and normalization technique, see [37]. Maximally degenerated, a collection of P1 (C)’s will appear. The examples considered in this article are of this type. The deformations appear as families of current algebras which are naturally defined over the moduli space of genus one curves (i.e. 1 Note that the maximal nilpotent Lie algebra of g is not formally rigid anymore, and its formal deformations are described in [4].
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of elliptic curves, or equivalently of complex one-dimensional tori) with two marked points. These deformations are associated to geometric degenerations of elliptic curves to singular cubic curves. The desingularization (or normalization) of their singularities will yield the projective line as normalization. We will end up with algebras related to the genus zero case. See Sect. 5 for the full geometric picture. The classical current (or affine) algebras appear as degenerations of elliptic two-point current algebras. From the opposite point of view the elliptic two-point current algebras are global deformations of the classical current algebra. But, as we show here, the structure of these algebras is not determined by the classical current algebras, despite the formal rigidity of the latter (if g is simple). The structure of the article is the following. In Sect. 2 we recall the different definitions of deformations of Lie algebras. In particular, we stress the fact that in the case of an infinite dimensional Lie algebra it is important to distinguish clearly the different notions, and indicate in which category we work: infinitesimal, formal, geometric, or analytic deformations. Also it is important to allow as a base of the deformations not only local algebras, but also global algebras. They correspond to geometric situations. The most simple global case is the case of the algebra of polynomials in one variable. Geometrically this corresponds to a deformation over the affine line. As already indicated above, in the formal case everything can be described in cohomological terms. To contrast the infinite dimensional case with the finite-dimensional one, we recall some results from their theory. In the finite-dimensional case for non-singular Lie algebras all notions of rigidity introduced above are equivalent and correspond to the fact that the cohomology space vanishes. The reason is that in this case the whole situation can be described within the frame of finite-dimensional algebraic geometry. In Sect. 3 we recall what is needed about the higher genus multi-point algebras of Krichever-Novikov type. The following algebras are introduced: the associative algebra of functions, the Lie algebras of vector fields and of currents, including their central extensions. For the currents the central extensions are the higher genus multi-point affine algebras. In Sect. 4 we construct geometric deformations of the standard current algebra by considering certain families of algebras for the genus one case (i.e. the elliptic curve case) and let the elliptic curve degenerate to a singular cubic. The two points, where poles are allowed, are the zero element of the elliptic curve (with respect to its group structure) and a 2-torsion point. In this way we obtain families parameterized over the affine line with the peculiar behaviour that every family is a global deformation of the classical current algebra, i.e. the classical current algebra is a special member, whereas all other members are mutually isomorphic but not isomorphic to the special element if the finite-dimensional Lie algebra is simple, see Theorem 4.10. Even if restricted to small open neighbourhoods of the point corresponding to the special element, these families are non-trivial, only infinitesimally and formally they are trivial. The construction can be extended to the centrally extended algebras, yielding global deformations of the affine algebra. The algebras are explicitly given by generators and structure equations and yield new examples of infinite dimensional algebras of current and affine Lie algebra type. In Sect. 5 we consider the geometric picture behind it. In particular, we identify those algebras we obtain over the nodal cubics (i.e. the cubic curves with one singular point with two tangent directions at this point). We explain the geometric reason why we obtain them. Depending whether the node will become the point where a pole is allowed
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or not, we obtain a three-point current algebra of genus zero or a certain subalgebra of the classical current algebra. In Sect. 6 we give the cohomology classes of our families of deformations. As it is known that these algebras are formally rigid in the simple case, it can be expected, also in the general case, that the cocycles will be coboundaries. We show this by direct calculations. In two appendices we calculate the cocycle defining the central extensions of our family. In the process of constructing our families we construct also families of the commutative and associative algebra of functions on Riemann surfaces with prescribed regularity. These are families deforming the algebra of Laurent polynomials C[z−1 , z]. This algebra is the coordinate algebra of the smooth affine curve P1 \ {0, ∞}. The cohomology corresponding to commutative and associative deformations (i.e. the Harrison cohomology) vanishes for such algebras. The algebra is infinitesimally and formally rigid. Nevertheless, we constructed nontrivial (geometrically) local families. Maybe, Kontsevich’s concept of semi-formal deformations [17] (related to filtrations of a certain type) might help to describe this situation better. Indeed, the families of associative algebras considered in this article are semi-formal deformations in his sense. 2. Some Generalities on Deformations of Lie Algebras 2.1. Intuitive description. Let us start with the intuitive definition of a Lie structure depending on a parameter t. Let L be a Lie algebra with Lie bracket µ = µ0 over a field K. A deformation of L is a one-parameter family Lt of Lie algebras (with the same underlying vector space) with the bracket µt = µ0 + tφ1 + t 2 φ2 + ...,
(2.1) 2
L, L) = where φi are L-valued (alternating) two-cochains, i.e. elements of HomK ( C 2 (L, L), and Lt is a Lie algebra for each t ∈ K (see [12, 21, 22]). In particular, we have L = L0 . Two deformations Lt and Lt are equivalent if there exists a linear automorphism ψˆ t = id + ψ1 t + ψ2 t 2 + ... of L, where ψi are linear maps over K, i.e. elements of C 1 (L, L), such that µt (x, y) = ψˆ t−1 (µt (ψˆ t (x), ψˆ t (y))).
(2.2)
These objects are related to Lie algebra cohomology. We recall the following definitions. A bilinear map ω : L ⊗ L → L is a Lie algebra two-cocycle with values in the adjoint representation L, if ω is alternating and fulfills 0 = d2 (ω)(x, y, z) := ω([x, y], z) − ω([x, z], y) + ω([y, z], x) −[x, ω(y, z)] + [y, ω(x, z)] − [z, ω(x, y)].
(2.3)
The vector space of two-cocycles is denoted by Z2 (L, L). A two-cocycle ω is a coboundary if there exists a linear map η : L → L such that ω(x, y) = (d1 η)(x, y) := η([x, y]) − [x, η(y)] − [η(x), y].
(2.4)
The vector space of coboundaries is denoted by B2 (L, L) and is a subspace of Z2 (L, L). The quotient space is the Lie algebra two-cohomology with values in the adjoint module, denoted by H2 (L, L), see e.g. [11, 2] for further details.
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Coming back to the deformation (2.1) we see that the Jacobi identity for the algebras Lt implies that the two-cochain φ1 is indeed a cocycle, i.e. it fulfills d2 φ1 = 0. If φ1 vanishes identically, the first nonvanishing φi will be a cocycle. If µt is an equivalent deformation (with cochains φi ) then φ1 − φ1 = d1 ψ1 .
(2.5)
Hence, every equivalence class of deformations defines uniquely an element of H2 (L, L). This class is called the differential of the deformation. The differential of a family which is equivalent to a trivial family will be the zero cohomology class. 2.2. Global deformations. The deformation Lt might be considered not only as a family of Lie algebras, but also as a Lie algebra over the algebra K[[t]] of formal power series over K. From this description a natural step is to allow K[[t1 , t2 , . . . , tk ]], or an arbitrary commutative algebra A over K with unit as the base of a deformation (see [6, 9]). In the following we will assume that A is a commutative algebra over K (where K is a field of characteristic zero) which admits an augmentation : A → K. This says that is a K-algebra homomorphism, e.g. (1A ) = 1. The ideal m := Ker is a maximal ideal of A. Vice versa, given a maximal ideal m of A with A/m ∼ = K, the natural quotient map defines an augmentation. If A is a finitely generated K-algebra over an algebraically closed field K then A/m ∼ = K is true for every maximal ideal m. Hence, in this case every such A admits at least one augmentation and all maximal ideals are coming from augmentations. Let us consider a Lie algebra L over the field K, a fixed augmentation of A, and m = Ker the associated maximal ideal. Definition 2.1 ([9]). A global deformation λ of L with base (A, m) or simply with base A, is a Lie A-algebra structure on the tensor product A ⊗K L with bracket [., .]λ such that ⊗id : A⊗L → K⊗L = L
(2.6)
is a Lie algebra homomorphism. Specifically, it means that for all a, b ∈ A and x, y ∈ L, (1) [a ⊗ x, b ⊗ y]λ = (ab ⊗ id )[1 ⊗ x, 1 ⊗ y]λ , (2) [., .]λ is skew-symmetric and satisfies the Jacobi identity, (3) ⊗ id ([1 ⊗ x, 1 ⊗ y]λ ) = 1 ⊗ [x, y]. By Condition (1) for describing a deformation it is enough to give the elements [1 ⊗ x, 1 ⊗ y]λ for all x, y ∈ L. If B = {zi }i∈J is a basis of L it follows from Condition (3) that the Lie product has the form [1⊗x, 1⊗y]λ = 1⊗[x, y] + i ai ⊗zi , (2.7) denotes a finite sum. Clearly, Condition (2) is with ai = ai (x, y) ∈ m, zi ∈ B. Here an additional condition which has to be satisfied. A deformation is called trivial if A ⊗K L carries the trivially extended Lie structure, i.e. (2.7) reads as [1 ⊗ x, 1 ⊗ y]λ = 1 ⊗ [x, y]. Two deformations of a Lie algebra L with the same base A are called equivalent if there exists a Lie algebra isomorphism between the two copies of A ⊗ L with the two Lie algebra structures, compatible with ⊗ id.
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We say that a deformation is local (in the algebraic sense) if A is a local K-algebra with unique maximal ideal mA . By assumption mA = Ker and A/mA ∼ = K. In case that in addition mA 2 = 0, the deformation is called infinitesimal, (see [6]). An important class of examples is given if A is the algebra of regular functions of an affine variety V , i.e. A = K[V ]. The algebra A is also called the coordinate algebra of the affine variety V . Let us assume that K is algebraically closed. It is known that every finitely generated (as ring) K-algebra which is reduced, i.e. which has no nilpotent elements, is the algebra of regular functions of a suitable affine variety. The variety might be reducible if A has zero-divisors. The maximal ideals mx of A correspond exactly to the points x ∈ X, and exactly to the set of augmentations x . Fixing an augmentation x0 means fixing a point x0 ∈ V . If A is a non-local ring, there will be different maximal ideals, hence also different augmentations. Let L be a K-vector space and assume that there exists a Lie A-algebra structure [., .]A on A ⊗K L. Given an augmentation : A → K with associated maximal ideal m = Ker , one obtains a Lie K-algebra structure L = (L, [., .] ) on the vector space L by passing to the quotient A/m . In this way the Lie algebra A ⊗K L gives a family of Lie algebra structures on the space L parameterized by the points of V . We obtain a map V
→
{ set of Lie algebra structures on the vector space L }.
(2.8)
We will denote both A = K[V ] and V as the base of the deformation. It is quite convenient to consider both (the algebraic and the geometric) pictures. Clearly, the Lie A-algebra A ⊗K L, i.e. the global family over A, is a deformation in the sense of Definition 2.1 with basis (A, mx ) for all points x ∈ V . If we choose a basis {Ta }a∈J of the Lie algebra L, the Lie structure in L is given by c } defined via the structure constants {Ca,b c [Ta , Tb ] = Ca,b Tc , a, b ∈ J. (2.9) c∈J
Here denotes that only a finite number of the summands will be different from 0. Assume that we have a deformation over A = K[V ] in the sense of Definition 2.1, where L lies above x0 . The elements 1 ⊗ Ta are now an A-basis of A ⊗ L and Eq. (2.7) can be written as c [1 ⊗ Ta , 1 ⊗ Tb ]x = Ca,b (x) 1 ⊗ Tc a, b ∈ J (2.10) c∈J
with algebraic functions c Ca,b (x) ∈ K[V ],
c c with (Ca,b (x) − Ca,b ) ∈ mx0 ,
c c or equivalently Ca,b (x0 ) = Ca,b .
(2.11)
Note that the range of the summation might be bigger than the original one. In this way we get back the intuitive picture of algebraically varying structure constants. The c (x ). structure constants of the Lie algebra Lx lying above the point x are given by Ca,b 1 A very important special case is a deformation over the affine line A . Here the corresponding algebra is A = K[A1 ] = K[t], the algebra of polynomials in one variable. For a deformation of the Lie algebra L = L0 over the affine line, the Lie structure Lα in the fiber over the point α ∈ K is given by considering the augmentation corresponding to the maximal ideal mα = (t − α).
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Remark. Despite the fact that we only consider deformations over affine varieties, we call this type of deformations global, as we also allow non-local algebras. Nevertheless, it might be useful to study deformations over a more general global basis, like projective varieties, schemes, complex manifolds, analytic spaces, etc. As in this article we are only interested in rigidity questions, and they are geometrically local, i.e. only the case of open neighbourhoods (e.g. local affine neighbourhoods) is of importance, see Definition 2.5, the given definition will be sufficient for us. For further reference we note Definition 2.2. Let A be another commutative algebra over K with a fixed augmentation : A → K, and let φ : A → A be an algebra homomorphism with φ(1) = 1 and ◦ φ = . If a deformation λ of L with base (A, Ker = m) is given, then the push-out λ = φ∗ λ is the deformation of L with base (A , Ker = m ), and Lie algebra structure [a1 ⊗A (a1 ⊗l1 ), a2 ⊗A (a2 ⊗l2 )]λ := a1 a2 ⊗A [a1 ⊗l1 , a2 ⊗l2 ]λ , (a1 , a2 ∈ A , a1 , a2 ∈ A, l1 , l2 ∈ L) on A ⊗ L = (A ⊗A A) ⊗ L = A ⊗A (A ⊗ L). Here A is regarded as an A-module with the structure aa = φ(a)a . ←−− 2.3. Formal deformations [6]. Let A be a complete local algebra over K, so A = lim n→∞ (A/mn ), where m is the maximal ideal of A. Furthermore, we will assume that A/m ∼ = K, and dim(mk /mk+1 ) < ∞ for all k. Definition 2.3. A formal deformation of L with base A is a Lie algebra structure on −− L = ← the completed tensor product A⊗ lim ((A/mn ) ⊗ L) such that n→∞
id : A⊗ L → K ⊗ L = L ⊗
(2.12)
is a Lie algebra homomorphism. If A = K[[t]], then a formal deformation of L with base A is the same as a formal 1-parameter deformation of L (see [12]). There is an analogous definition for equivalence of deformations parameterized by a complete local algebra. 2.4. Isomorphy of finite-dimensional Lie algebras and GL(n) orbits. In this subsection let L be a finite-dimensional Lie algebra of dimension n over K. In a given basis c } denote the structure constants of a fixed Lie algebra µ by {Ca,b a,b,c=1,... ,n (see (2.9)). Obviously, the relations on the structure constants c c + Cb,a = 0, Ca,b n
l d l d Cl,c + Cb,c Cl,a Ca,b
a, b, c = 1, . . . , n, l d + Cc,a Cl,b = 0, a, b, c, d = 1, . . . , n,
(2.13)
l=1
are algebraic equations, and the vanishing set Lalgn in KN , N = n3 , of the ideal generated by them “parameterizes” the possible Lie algebra structures on the n-dimensional vector space V . Strictly speaking, it is more appropriate to talk about the scheme, as
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one should better consider the not necessarily reduced structure on Lalgn , see Rauch [24]. Furthermore, as µ is a bilinear map V × V → V , the structure constants might be considered more canonically as elements of V ∗ ⊗ V ∗ ⊗ V with V ∗ denoting the dual space of V . Given a linear automorphism ∈ GL(V ), it will define an action on V ∗ ⊗ V ∗ ⊗ V by ( µ)(x, y) = (µ( −1 (x), −1 (y))),
(2.14)
which respects the conditions defining a Lie algebra structure. The Lie algebras (V , µ) and (V , µ ) will be isomorphic iff µ and µ are in the same orbit under the GL(V ) action. On the level of structure constants, i.e. after choosing a basis in V , we obtain a GL(n) action on Lalgn . In this way the isomorphy classes of Lie algebras of dimension n correspond exactly to the GL(n) orbits of Lalgn . For further reference we note that a finite-dimensional Lie algebra L is called nonsingular if the corresponding point µ in the scheme Lalgn is a non-singular point in the sense of algebraic geometry. 2.5. Rigidity. Intuitively, rigidity of a Lie algebra L means that we cannot deform the Lie algebra. Or, formulated differently, given a family of Lie algebras containing L as the special element L0 , any element Lt in the family “nearby” will be isomorphic to L0 . Of course, the definition depends on the category in which deformations are considered. As “nearby” is already encoded in the infinitesimal and formal setting, we can define: Definition 2.4. (a) A Lie algebra L is infinitesimally rigid iff every infinitesimal deformation of it is equivalent to the trivial deformation. (b) A Lie algebra L is formally rigid iff every formal deformation of it is equivalent to the trivial deformation. For the global deformation such a condition would be too much to require as the following example shows. Example. Let L be any non-abelian Lie algebra with Lie bracket [., .]. We define a family of Lie algebras Lt over the algebra K[t] (i.e. geometrically over the affine line K) by taking as Lie bracket the bracket [x, y]t := (1 − t)[x, y]. If we set t = 0 we obtain back our Lie algebra L. Moreover, as long as t = 1 the algebras Lt are isomorphic to L, but L1 as abelian Lie algebra will be non-isomorphic. Hence such a family will never be the trivial family. But if we restrict the family to the (Zariski) open subset K \ {1}, of the base K, we obtain a trivial family. In the pure algebraic setting we can apply Zariski topology. The Zariski topology (Z-topology) of a ring R is a topology on the set of prime ideals, yielding a topological space Spec(R). Instead of considering the general case, we will restrict ourselves to the situation which is of interest here. Let K be an algebraically closed field of characteristic zero and A a finitely generated reduced K-algebra. Recall that this corresponds geometrically to the case of affine varieties V as base. A subset W of V is called Z-closed if it is the vanishing set of an ideal in A = K[V ]. The set W itself will be an affine variety. A subset U of V is called Z-open if it is the complement of a closed set. A basis of the open set of the topology is given by Z-open affine varieties. These basis sets are open
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sets in V and they themselves are affine (this says they are the vanishing sets of an ideal in a suitable affine space), see [14] for this and further results. In particular, given an arbitrary Z-open subset U containing the point x ∈ V , we can always find an open subset W of U which itself is affine. For simplicity (and in accordance with the general picture) let us use Spec(A) to denote the variety V and let us identify the (closed) points in V with the maximal ideals. Definition 2.5. A Lie algebra L is algebraic-geometrically rigid or just geometrically rigid if for every deformation of L in the sense of Definition 2.1 with base (A, m), where A is a finitely generated reduced algebra over the algebraically closed field K and m a maximal ideal, there exists a Zariski open affine neighbourhood Spec(B) of the point m in Spec(A) such that the restricted family over (B, m) is equivalent to the trivial family. Note that the restriction given by the geometric picture Spec(B) → Spec(A) corresponds exactly to the push-out of the deformation given by the induced K-algebra homomorphism A → B. Note also that with a slight abuse of notation we understand m in (B, m) being the ideal generated by the image in B of the ideal m of A. In the case if the base field is C (or R) and the base of the deformation is a finitedimensional analytic manifold M with a chosen point x0 ∈ M, we consider also the usual topology on M. In particular, we can talk about open subsets of M containing x0 . Definition 2.6. A Lie algebra L over C (or R) is analytically rigid if for every family over a finite-dimensional analytic manifold (M, x0 ) with base point x0 and special fiber L∼ = Lx0 over the point x0 ∈ M, there is an open neighbourhood U of x0 , such that the restriction of this family is equivalent to the trivial family. Obviously, as the analytic rigidity is a (geometrically) local condition, it is enough to establish rigidity by considering families over Ck (resp. Rk ). There are other definitions of rigidity by considering differentiable manifolds or analytic spaces as base. Moreover, we could incorporate infinite dimensional manifolds or non-Noetherian algebras as base of the deformations. But for the families discussed in this article we do not need them, hence we will not discuss them here. Already in the case that the Lie algebra is infinite dimensional and the base is finite-dimensional, interesting examples appear. It can be expected that the situation will become even more intricate by allowing infinite dimensional bases. To our knowledge not much is known in this direction, whereas the topic is surely interesting. We intend to come back to it in the future. In case that the Lie algebra is finite-dimensional there is another important definition of rigidity. Definition 2.7 ([12, 22]). Let L be a Lie algebra of dimension n corresponding to the c } in the space of structure constants. The algebra L is called rigid (in point µ = {Ca,b c } under the GL(n) action is Zariski the orbit sense) or just rigid if the orbit of {Ca,b open in Lalgn . Given a finite-dimensional Lie algebra L which is rigid (in the orbit sense) and a deformation over the base (V , x0 ), then the algebras Lx will be isomorphic to Lx0 = L for any x in a Zariski open neighbourhood U of x0 . Hence, (orbital) rigidity implies geometric (and if K is equal R or C also analytic) rigidity. See Theorem 2.9 and Theorem 2.12 for partial inversions of this statement.
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2.6. Rigidity and cohomology. As explained in Sect. 2.1 there is a close connection between deformations of Lie algebras and the Lie algebra two-cohomology H2 (L, L) with values in L. The following is obvious. Proposition 2.8. Let L be an arbitrary Lie algebra over K, then L is infinitesimally rigid iff H2 (L, L) = 0. Theorem 2.9 ([12, 21, 22]). Let L be a finite-dimensional Lie algebra. If H2 (L, L) = 0 then the Lie algebra L is rigid in all the senses introduced above. In particular, as infinitesimal rigidity is equivalent to H2 (L, L) = 0, it already implies in finite dimension rigidity in the formal, geometric, (and analytic) sense. This is definitely not true in infinite dimension, as we showed in [10] by exhibiting a deformation over the affine line C[t] of the Witt algebra which is non-trivial in every neighbourhood of t = 0, despite the fact that the Witt algebra is infinitesimally and formally rigid. In Sect. 4 we will construct deformations of the current algebras g associated to finite dimensional complex Lie algebras g which are neither geometrically nor analytically rigid despite that for g simple the Lie algebra g is formally rigid [20]. Nevertheless, also in infinite dimensions we have the following theorem. Theorem 2.10 ([6]). Let L be an arbitrary Lie algebra. If H2 (L, L) = 0 then L is formally rigid. The theorem is a corollary of the result of Fialowski [6] on the existence of versal formal families which will be recalled in the next subsection. For completeness and to contrast this with the infinite dimensional case let us quote the following results for the finite-dimensional case. Theorem 2.11 ([24]). Let L be a finite-dimensional Lie algebra corresponding to the point µ ∈ Lalgn then: (a) The Zariski tangent space of the scheme Lalgn at the point µ can be naturally identified with Z2 (L, L). (b) The Zariski tangent space of the GL(n) orbit of µ (considered as reduced scheme) can be naturally identified with B2 (L, L). From Theorem 2.11 and Theorem 2.9, and the fact that H2 (L, L) = 0 implies that µ is a non-singular point [12, 22], the following is a consequence. Theorem 2.12. Let L be a finite-dimensional Lie algebra. Then H2 (L, L) = 0 iff L is rigid and is a nonsingular Lie algebra. Moreover, in this case all the above definitions of rigidity coincide. Richardson [25] gave an example of a finite-dimensional Lie algebra with H2 (L, L) = 0, which is nevertheless rigid in the orbit sense. Clearly, the µ corresponding to the Lie structure of his L is a singular point in Lalgn . 2.7. Universal and versal deformations. As explained above and will be shown later in this article, the strong relation between cohomology spaces and deformations over nonlocal rings breaks down in infinite dimension. It might be considered as an astonishing result that a tight connection still exists on the formal level.
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Proposition 2.13 ([9]). Assume that dim H2 (L, L) < ∞, then there exists a universal infinitesimal deformation ηL of the Lie algebra L with base B = K ⊕ H2 (L, L)∗ , where the second summand is the dual of H2 (L, L) equipped with the zero multiplication, i.e. (α1 , h1 ) · (α2 , h2 ) = (α1 α2 , α1 h2 + α2 h1 ). This means that for every infinitesimal deformation λ of the Lie algebra L with finite dimensional base A, there exists a unique homomorphism φ : K ⊕ H2 (L, L)∗ → A such that λ is equivalent to the push-out φ∗ ηL . Although in general it is impossible to construct a universal formal deformation, there is a so-called versal element. Definition 2.14 ([5, 6]). A formal deformation η of L parameterized by a complete local algebra (B, mB ) is called versal if for every deformation λ, parameterized by a complete local algebra (A, mA ), there is a morphism f : B → A such that 1) The push-out f∗ η is equivalent to λ. 2) If A satisfies mA 2 = 0, then f is unique. Theorem 2.15 ([6, 9],Thm. 4.6). Assume that dim H2 (L, L) < ∞. (a) There exists a versal formal deformation of L. (b) The base of the versal formal deformation is formally embedded into H2 (L, L), i.e. it can be described in H2 (L, L) by a finite system of formal equations. Hence if H2 (L, L) = 0, every formal deformation will be equivalent to the trivial one, see Theorem 2.10.
2.8. Deformations of commutative algebras and Harrison cohomology. In this article we will deform current algebras by deforming associative and commutative algebras in a geometric way. The corresponding cohomology theory of such deformations is the Harrison cohomology [15]. Let A be an associative and commutative algebra over K. We only need here the space 2 HH arr (A, A). Recall its definition. The two-cocycles are bilinear maps F : A×A → A such that F (a, b) = F (b, a), a, b ∈ A, δ2 F (a, b, c) := aF (b, c) − F (ab, c) + F (a, bc) − F (a, b)c = 0,
a, b, c ∈ A. (2.15)
A two-cycle is a coboundary if there exists a linear map φ : A → A such that F = δ1 φ(a, b) = aφ(b) − φ(ab) + φ(a)b,
a, b ∈ A.
(2.16)
2 Note that HH arr (A, A) will be a subspace of the Hochschild cohomology space 2 HH och (A, A).
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3. Krichever-Novikov Algebras 3.1. The algebras with their almost-grading. Algebras of Krichever-Novikov type are generalizations of the Virasoro algebra, current algebras and all their related algebras. Let M be a compact Riemann surface of genus g, or in terms of algebraic geometry, a smooth projective curve over C. Let N, K ∈ N with N ≥ 2 and 1 ≤ K < N . Fix I = (P1 , . . . , PK ),
and
O = (Q1 , . . . , QN−K )
disjoint ordered tuples of distinct points (“marked points”, “punctures”) on the curve. In particular, we assume Pi = Qj for every pair (i, j ). The points in I are called the in-points, the points in O the out-points. Sometimes we consider I and O simply as sets and denote A = I ∪ O as a set. Here we will need the following algebras. Let A be the associative algebra of those meromorphic functions on M which are holomorphic outside the set of points A with point-wise multiplication. Let L be the Lie algebra of meromorphic vector fields which are holomorphic outside of A with the usual Lie bracket of vector fields. The algebra L is called the vector field algebra of Krichever-Novikov type. They were introduced and their structure was studied by Krichever and Novikov [18]. The corresponding generalization to the multi-point case was done in [26–29]. Obviously, both A and L are infinite dimensional algebras. Furthermore we will need the higher-genus, multi-point current algebra of KricheverNovikov type [31, 34]. We start with g being a complex finite-dimensional Lie algebra and endow the tensor product G = g ⊗C A with the Lie bracket [x ⊗ f, y ⊗ g] = [x, y] ⊗ f · g,
x, y ∈ g,
f, g ∈ A.
(3.1)
The algebra G is the higher genus current algebra. It is an infinite dimensional Lie algebra and might be considered as the Lie algebra of g-valued meromorphic functions on the Riemann surface without poles outside of A. The classical genus zero and N = 2 point case is given by the geometric data M = P1 (C) = S 2 ,
I = {z = 0},
O = {z = ∞}.
(3.2)
In this case the algebras are the well-known algebras of Conformal Field Theory (CFT). For the function algebra we obtain A = C[z−1 , z], the algebra of Laurent polynomials. d The vector field algebra L is the Witt algebra generated by ln = zn+1 dz , n ∈ Z with Lie bracket [ln , lm ] = (m − n)ln+m , and the current algebra G is the standard current algebra g = g ⊗ C[z−1 , z] with Lie bracket [x ⊗ zn , y ⊗ zm ] = [x, y] ⊗ zn+m
x, y,
n, m ∈ Z.
(3.3)
For infinite dimensional algebras, modules and their representation theory a graded structure is usually important to obtain structure results. In fact, to define certain types of representations which are fundamental e.g. in CFT, like the highest weight representations, one uses a grading. In the classical situation the algebras are obviously graded by taking as degree deg ln := n and deg x ⊗ zn := n. For higher genus there is usually no grading. But it was observed by Krichever and Novikov in the two-point case [18] that a weaker concept, an almost-graded structure, will be enough to develop an interesting theory of representations (Verma modules, etc.).
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Definition 3.1. Let A be an (associative or Lie) algebra admitting a direct decompo sition as vector space A = A . The algebra A is called an almost-graded n n∈Z algebra if (1) dim An < ∞ and (2) there are integer constants R and S such that An · Am
⊆
n+m+S
Ah ,
∀n, m ∈ Z .
(3.4)
h=n+m+R
The elements of An are called homogeneous elements of degree n. By exhibiting a special basis, for the multi-point situation such an almost-grading was introduced in [26–29]. Essentially, this is done by fixing the zero order of the basis elements at the points in I and O in a complementary way to make them unique. In this way we obtain e.g. for the function algebra (resp. for the vector field algebra) basis elements An,p (resp. en,p ) with n ∈ Z and p = 1, . . . , K. As definition for the degree we take (x ∈ g) deg(en,p ) := deg(An,p ) := deg(x ⊗ An,p ) := n.
(3.5)
In the following we will give an explicit description of the basis elements for those genus zero and one situations we need. Hence, we will not recall their general definition but only refer to the above quoted articles. Proposition 3.2 ([26, 29]). With respect to the grading introduced by (3.5) the algebras L, A, and G are almost-graded. The almost-grading depends on the splitting A = I ∪O. 3.2. Central extensions. In the construction of infinite dimensional representations of these algebras with certain desired properties (generated by a vacuum, irreducibility, unitarity, etc.) one is typically forced to “regularize” a “naive” action to make it welldefined. Important examples in CFT are the fermionic Fock space representations which are constructed by taking semi-infinite forms of a fixed weight. From the mathematical point of view, with the help of a prescribed procedure one modifies the action to make it well-defined, but on the other hand, accepting that the modified action in compensation will be only a projective Lie action. Such projective actions are honest Lie actions for a suitable centrally extended algebra. In the classical case they are well-known. The unique non-trivial (up to equivalence and rescaling) central extension of the Witt algebra is the Virasoro algebra. For the current algebra g ⊗ C[z−1 , z] if g is a simple Lie algebra with Cartan-Killing form β, it is the corresponding affine Lie algebra g (or, untwisted affine Kac-Moody algebra): −n · t, [x ⊗ zn , y ⊗ zm ] = [x, y] ⊗ zn+m − β(x, y) · n · δm [t, g] = 0, x, y ∈ g, n, m ∈ Z.
(3.6)
For the extension to higher genus and many points the objects have to be “geometrized”. First recall that for a Lie algebra V central extensions are classified (up to equivalence) by the second Lie algebra cohomology H2 (V, C) of V with values in the trivial module C. A bilinear form ψ : V × V → C is called a Lie algebra 2-cocycle iff ψ is antisymmetric and fulfills the cocycle condition 0 = d2 ψ(x, y, z) := ψ([x, y], z) + ψ([y, z], x) + ψ([z, x], y).
(3.7)
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To obtain central extensions of G = g ⊗ A we start with g being an arbitrary finitedimensional Lie algebra and β a symmetric, invariant, bilinear form on it (not necessarily non-degenerate). Invariance means that we have β([x, y], z) = β(x, [y, z]) for all = C ⊕ G as vector space and introduce the bracket x, y, z ∈ g. We set G 1 = 0 . (3.8) [x ⊗ f , y ⊗ g] = [x, y] ⊗ (f g) + β(x, y) f dg · t, [ t, G] 2πi CS Here we used for short x ⊗ f := (0, x ⊗ f ), t := (1, 0) and CS denotes a cycle separating the points in I from the points in O.
Proposition 3.3 ([33]). The term β(x, y) 2π1 i CS f dg is a Lie algebra 2-cocycle of G with structure (3.8) is a with values in the trivial module C. Hence, the vector space G Lie algebra which defines a central extension of G. These algebras are called higher genus (multi-point) affine algebras (of Krichever-Novikov type). In the classical situation (3.2) we obtain back (3.6). In the following we will use the term classical current algebra to denote the current algebra (3.6). In the new terminology it is a genus zero and two-point current algebra. Note that the cocycle can be calculated as β(x, y) ·
1 2π i
f dg = β(x, y) · CS
K
resPk (f dg) = −β(x, y) ·
k=1
N−K
resQl (f dg).
l=1
(3.9) As in general a non-vanishing β is not unique (up to rescaling), the central extension will depend on it. Even in the case when g is a simple Lie algebra, which implies G that there is essentially only one non-vanishing form β, the Cartan-Killing form, for the higher genus or/and multi-point situation H2 (G, C) will be more than one-dimensional (e.g. take another path of integration which is not homologous to CS ). But the following is shown in [33]. Theorem 3.4. Let g be a simple finite-dimensional Lie algebra, then up to equivalence and multiplication with a scalar there is a unique non-trivial 2-cohomology class for the current algebra G = g ⊗ A which has a “local” representative, i.e. which allows defined by the local repreto extend the almost-grading of G to the central extension G sentative by assigning to the central element t a degree. This extension is given by (3.8) with β being the Cartan-Killing form. Also, in [33] a thorough treatment can be found for the case when g is semi-simple or even reductive. Corresponding uniqueness results for almost-graded central extensions of A and L are shown in [32]. 4. Current Algebras for the Elliptic Curve Case In the following we will construct global deformations of the classical current algebra g = g ⊗ C[z−1 , z] and its central extension g. First we construct a family of associative algebras which contain the algebra of Laurent polynomials C[z−1 , z] as special element. The deformation family for the current algebra will be obtained by tensoring g with this
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family. These omations will be of geometric origin. More precisely, each non-special element in the family will be a current algebra of Krichever-Novikov type for the genus one (i.e. the elliptic) case. As a side effect we obtain new examples of explicitly given infinite dimensional algebras of current and affine Lie algebra type. The construction of these families will not make any assumption about the finite-dimensional Lie algebra. Only later we will require g to be a simple Lie algebra to contrast the existence of these global deformations which are (geometrically) locally not equivalent to trivial deformations, with the formal rigidity of g [20]. 4.1. The family of elliptic curves. For the convenience of the reader we will recall the geometric picture from [10]. As we have geometric degenerations in mind, it is more convenient to pass from the complex analytic picture (i.e. the language of Riemann surfaces) to the algebraic geometric picture (i.e. the language of curves). Every compact Riemann surface of genus one corresponds to an elliptic curve in the projective plane. Recall that the elliptic curves can be given as sets of solutions of the polynomial equation Y 2 Z = 4X3 − g2 XZ 2 − g3 Z 3 ,
g2 , g3 ∈ C,
with := g2 3 − 27g3 2 = 0. (4.1)
Here g2 and g3 are parameterizing the individual curve and the condition = 0 assures that the curve will be nonsingular. Instead of (4.1) we can use the description Y 2 Z = 4(X − e1 Z)(X − e2 Z)(X − e3 Z)
(4.2)
with e1 + e2 + e3 = 0,
and = 16(e1 − e2 )2 (e1 − e3 )2 (e2 − e3 )2 = 0.
(4.3)
These presentations are related via g2 = −4(e1 e2 + e1 e3 + e2 e3 ),
g3 = 4(e1 e2 e3 ).
(4.4)
The elliptic modular parameter classifying elliptic curves up to isomorphy is given as j = 1728
g23 .
(4.5)
We set B := {(e1 , e2 , e3 ) ∈ C3 | e1 + e2 + e3 = 0,
ei = ej for i = j }.
(4.6)
Inside the product B × P2 we consider the family of elliptic curves E over B defined via (4.2), with its obvious projection E → B. The family can be extended to := {e1 , e2 , e3 ) ∈ C3 | e1 + e2 + e3 = 0}. B
(4.7)
\ B are singular cubic curves. Resolving the linear relation in B via The fibers above B e3 = −(e1 + e2 ) we obtain a family over C2 . Consider the following complex lines in C2 : Ds := {(e1 , e2 ) ∈ C2 | e2 = s · e1 },
s ∈ C,
D∞ := {(0, e2 ) ∈ C2 }. (4.8)
Set also Ds∗ = Ds \ {(0, 0)}
(4.9)
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for the punctured line. Now B∼ = C2 \ (D1 ∪ D−1/2 ∪ D−2 ).
(4.10)
∗ Note that above D1∗ we have e1 = e2 = e3 , above D−1/2 we have e2 = e3 = e1 , and ∗ above D−2 we have e1 = e3 = e2 . In all these cases we obtain a nodal cubic. Every nodal cubic EN can be given as
Y 2 Z = 4(X − eZ)2 (X + 2eZ),
(4.11)
where e denotes the value of the coinciding ei = ej (−2e is then necessarily the remaining one). The singular point is the point (e : 0 : 1). It is a node. Above the unique common intersection point (0, 0) of all Ds there is the cuspidal cubic EC : Y 2 Z = 4X3 .
(4.12)
The singular point is (0 : 0 : 1). In both cases the complex projective line is the desingularisation of the singular curve. In all cases (non-singular or singular) the point ∞ = (0 : 1 : 0) lies on the curves. It is the only intersection with the line at infinity, and is a non-singular point. In passing to the affine plane in the following we will loose nothing. The affine curve will be given as the solution set of Y 2 = 4(X − e1 )(X − e2 )(X − e3 ).
(4.13)
For the curves above the points in Ds∗ we calculate e2 = se1 and e3 = −(1 + s)e1 (resp. e3 = −e2 if s = ∞). Due to the homogeneity, the modular parameter j for the curves above Ds∗ will be constant along the line. In particular, the curves in the family lying above Ds∗ will be isomorphic. Their modular parameter calculates to j (s) = 1728
4(1 + s + s 2 )3 , (1 − s)2 (2 + s)2 (1 + 2s)2
j (∞) = 1728.
(4.14)
4.2. The family of current algebras. First we define a family of function algebras A on these elliptic curves. We introduce the points where poles are allowed. For our purpose it is enough to consider two marked points. We will always put one marking to ∞ = (0 : 1 : 0) and the other one to the point with the affine coordinate (e1 , 0). These ∼ markings define two sections of the family E over B = C2 . With respect to the group structure on the elliptic curve given by ∞ as the neutral element (the first marking), the second marking chooses a two-torsion point. All other choices of two-torsion points will yield isomorphic situations. ∗ ∗ ) Proposition 4.1. For every elliptic curve E(e1 ,e2 ) over (e1 , e2 ) ∈ C2 \(D1∗ ∪D−1/2 ∪D−2 the associative algebra A(e1 ,e2 ) of functions on E(e1 ,e2 ) has a basis {An , n ∈ Z} such that the algebra structure is given as for n or m even, An+m , An · Am = An+m + 3e1 An+m−2 (4.15) +(e1 − e2 )(2e1 + e2 )An+m−4 , for n, m odd.
By setting deg(An ) := n, we obtain an almost-grading.
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Proof. In [30] a vector space basis of the meromorphic forms of arbitrary integer weights was given. For weight 0, i.e. the Krichever-Novikov function algebra A, the basis specializes to A2k := (X − e1 )k ,
A2k+1 := 1/2Y (X − e1 )k−1
k ∈ Z.
(4.16)
Taking the product, for n or m even the result is immediate. In the case when both n and m are odd, one replaces Y 2 by 4(X − e1 )(X − e2 )(X − e3 ) in the product and uses e1 + e2 + e3 = 0. The almost-grading is obvious. The algebras in Proposition 4.1 defined with the structure (4.15) make sense also for the points (e1 , e2 ) ∈ D1 ∪ D−1/2 ∪ D−2 . Altogether this defines a two-dimensional family of algebras parameterized over the affine plane C2 , or described algebraically, over the polynomial algebra C[e1 , e2 ]. For (e1 , e2 ) = (0, 0) we obtain the algebra of Laurent polynomials, if we make the identification An = zn . By tensoring this family with g we get the following. Theorem 4.2. Let g be a finite-dimensional Lie algebra, and V a vector space with basis {An | n ∈ Z}. Then for given complex values e1 , e2 the vector space g ⊗ V carries the structure of a Lie algebra given by for n or m even, [x, y] ⊗ An+m , [x ⊗ An , y ⊗ Am ] = [x, y] ⊗ An+m + 3e1 [x, y] ⊗ An+m−2 +(e1 −e2 )(2e1 +e2 )[x, y] ⊗ An+m−4 , for n, m odd. (4.17) Here x and y are elements of g. In particular, these algebras define a two-parameter family of deformations of the current algebra g, such that g corresponds to the point (0, 0) and the algebra over (e1 , e2 ) ∈ B corresponds to the elliptic affine algebra G (e1 ,e2 ) fixed by the geometric data. ˆ But first we study the Later we will identify the algebras over the other points of B. family of algebras obtained by taking as base variety the (affine) line Ds (for any s). Corollary 4.3. For every s ∈ C ∪ {∞} the families of Theorem 4.2 define by restriction s one-dimensional families of Lie algebras G (e) which are deformations of the classical current algebra g (corresponding to e = 0) over the affine line C, resp. the algebra C[e]. For s = ∞ the family is given by for n or m even, [x, y] ⊗ An+m , [x ⊗ An , y ⊗ Am ] = [x, y] ⊗ An+m + 3e[x, y] ⊗ An+m−2 +e2 (1 − s)(2 + s)[x, y] ⊗ An+m−4 , for n, m odd. (4.18) For s = ∞ the family is given by [x, y] ⊗ An+m , for n or m even, [x ⊗ An , y ⊗ Am ] = [x, y] ⊗ An+m − e[x, y] ⊗ An+m−4 , for n, m odd. (4.19) s
In the case when s = 1, −1/2 or 2, the algebras G (e) are elliptic current algebras. In s s any case for fixed s we have G (e) ∼ = G (e ) as long as both e, e = 0.
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Proof. If s = ∞ then e2 = se1 and (4.17) reduces to (4.18) if we set e := e1 . For s = ∞ we have e1 = 0 and obtain (4.19) if we set e := e22 . If we rescale the basis elements √ = 0 always the algebra with e = 1 in A∗n = ( e)−n An (for s = ∞), we obtain for e √ our structure equations. For s = ∞ a rescaling ( 4 e)−n An will do the same (for e = 0). Hence we see that for fixed s in all cases the algebras will be isomorphic above every point in Ds , as long as we are not above (0, 0). Note that these algebras are new explicitly given examples of infinite dimensional Lie algebras. s s The generic isomorphy class of the family G (e) can be given by taking G (1) . To avoid s cumbersome notation we often denote this isomorphy type just by G . But we have to s keep in mind that by Proposition 4.9 further down the algebra G will not be isomorphic s to the special element G (0) = g. Clearly, if we do the same kind of restrictions to Ds for the families of associative algebras A(e1 ,e2 ) , we obtain similar one-dimensional algebraic families of commutative and associative algebras which deform the algebra of Laurent polynomials. We will denote these families by As(e) . Again for fixed s all algebras over e = 0 will be isomorphic and we will denote this isomorphy type simply by As . Proposition 4.8 will show that they are not isomorphic to the special element, the algebra of Laurent polynomial. 4.3. The three point case for P1 . There is another geometric family of algebras around. Its geometric picture is the three point situation for the Riemann sphere (the projective line). Because we will need these algebras later on anyhow, we will also give this family. The geometric situation is M = P1 (C), I = {α, −α} and O = {∞}, α = 0. Let us denote the corresponding Krichever-Novikov function algebra by V(α) . Proposition 4.4. For every α ∈ C∗ the algebra V(α) has a basis {An | n ∈ Z} such that its structure is given by An+m , for n or m even, An · Am = (4.20) 2 An+m + α An+m−2 , for n, m odd. By setting deg(An ) := n the algebra becomes an almost-graded algebra. Proof. In [30] it was shown that a vector space basis of V(α) is given by A2k := (z − α)k (z + α)k ,
A2k+1 := z(z − α)k (z + α)k ,
k ∈ Z. (4.21)
Here z is the quasi-global coordinate on P1 (C). The algebra structure follows now by direct calculations. Again, the algebra structure also makes sense for α = 0. In this case one obtains the algebra of Laurent polynomials. For the corresponding current algebra Y(α) := g ⊗ V(α) we obtain the following. Theorem 4.5. There exists a one-dimensional family of current algebras Y(α) with Lie structure [x, y] ⊗ An+m , for n or m even, [x ⊗ An , y ⊗ Am ] = [x, y] ⊗ A 2 n+m + α [x, y] ⊗ An+m−2 , for n, m odd, (4.22)
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which is a deformation of the classical current algebra g. Moreover, Y(α) is equivalent −2
1
to the families G (e) and G (e) . In Sect. 5 we will give a geometric explanation for the latter identification. 4.4. Families of the centrally extended algebras. The above family of current algebras s G (e1 ,e2 ) , resp. the one-dimensional families G (e) , can be centrally extended to a family s , i. e. to families of higher genus affine Lie algebras. This (e1 ,e2 )) , resp. G of algebras G (e) can be achieved by using the defining equation (3.8). Theorem 4.6. Assume we have a finite-dimensional Lie algebra g and a symmetric invariant bilinear form β on g. For the family G (e1 ,e2 ) of current algebras (4.17) containing the classical current algebra g as special element, a family of almost-graded (e1 ,e2 ) , i.e. a family of higher genus affine Lie algebras containing central extensions G the classical affine Lie algebra g (with respect to the form β) as special element, is given via the geometric two-cocycle (3.8). It calculates as 1 γ (x ⊗ An , y ⊗ Am ) = p(e1 , e2 ) · β(x, y) · An dAm (4.23) 2π i CS with 1 2πi
CS
−n , −nδm n, m even, 0, n, m different parity, An dAm = −n + 3e (−n + 1)δ −n+2 −nδ 1 m m +(e − e )(2e + e )(−n + 2)δ −n+4 , n, m odd. 1 2 1 2 m (4.24)
Here p(e1 , e2 ) is an arbitrary polynomial in the variables e1 and e2 . The proof involves residue calculus (see Eq. (3.9)) and will be postponed to an appendix. As the cocycle values (4.24) vanish if 0 ≤ n + m ≤ 4, the centrally extended algebras are almost-graded by setting deg t := 0 (or any other fixed value). Clearly, for e1 = e2 = 0 we obtain the classical affine algebra g. By restricting this two-dimensional family to the lines Ds we get one-dimensional families. For s = ∞ this amounts to replacing in the last term (e1 − e2 )(2e1 + e2 ) by e12 (1 − s)(2 + s). In particular, over D1 and D−2 this form will vanish for n and m both odd. By the uniqueness result for almost-graded central extensions (Theorem 3.4) in the case that g is simple we obtain Corollary 4.7. If g is a finite-dimensional simple Lie algebra then β is necessarily the Cartan-Killing form. Furthermore, (1) every almost-graded central extension for the family of current algebras G (e1 ,e2 ) is (e1 ,e2 ) (4.23). given up to fibrewise equivalence by G (e1 ,e2 ) with p = p(e1 , e2 ) ≡ const = 0 is up to fibrewise equivalence (2) In particular, G by the unique nontrivial almost-graded central extension of G (e1 ,e2 ) . Based on the classification results of [33] in the semi-simple case, the first part of Corol lary 4.7 remains true if one replaces the factor p(e1 , e2 )β(x, y) by K i=1 pi (e1 , e2 ) βi (x, y), where again pi are polynomials, and βi are the Cartan-Killing forms of the K simple summands of g.A similar statement is true for the reductive case, now additionally requiring the defining cocycle to be L-invariant (see [33] for the definition).
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4.5. Isomorphy question. In this subsection we will show that the current algebras G are not isomorphic to the classical current algebra g if g is semi-simple. Hence, the families introduced are non-trivial algebraic-geometric deformations for the (formally) rigid classical current algebra g (in the case when g is simple). Proposition 4.8. The algebras As are not isomorphic to the algebra of Laurent polynomials. Proof. The involved algebras are the algebras of meromorphic functions on projective curves with at least one point removed. Such curves are affine curves and the algebras are the affine coordinate algebras (i.e. the algebras of regular functions) of these curves. The isomorphy class of the coordinate algebra is uniquely given by the isomorphy class of the affine curve, e.g. see [14]. Any algebra As for s = 1, −2, −1/2 corresponds to an elliptic curve with two points removed, both algebras A1 and A−2 correspond to the projective line with three points removed, and A−1/2 corresponds to the nodal cubic with two (nonsingular) points removed. But the algebra of Laurent polynomials corresponds to the projective line with two points removed. The affine curves are not isomorphic as can easily be seen from the fact that their fundamental groups are different. Hence, also the algebras are not isomorphic. Proposition 4.9. Let g be a semi-simple finite-dimensional Lie algebra, and A and B two associative, commutative algebras (with units). If the current algebras g ⊗ A and g ⊗ B are isomorphic as Lie algebras then A and B are isomorphic as associative algebras. Proof. Let g be a semi-simple finite-dimensional Lie algebra, and P : g ⊗ A →: g ⊗ B a Lie isomorphism of the current algebras. The Lie algebra g admits a sl(2) subalgebra and hence also g ⊗ A admits a sl(2) ⊗ A subalgebra. By restriction we obtain a Lie isomorphism P : sl(2) ⊗ A → P (sl(2) ⊗ A). Via sl(2) ∼ = sl(2) ⊗ 1A (1A the unit in A), sl(2) is a subalgebra of sl(2) ⊗ A. Denote by h, e, f the standard generators of sl(2), i.e. 1 0 01 00 h= , e= ,f = , [h, e] = 2e, [h, f ] = −2f, [e, f ] = h. 0 −1 00 10 (4.25) The image P (sl(2)⊗1A ) is isomorphic to sl(2). In particular, P (h⊗1A ) will be mapped to a basis of its Cartan subalgebra. After applying an inner automorphism, we can assume that P (h ⊗ 1A ) = h ⊗ a1 with a1 ∈ B, a1 = 0. Let P (e ⊗ 1A ) = h ⊗ b1 + e ⊗ b2 + f ⊗ b3 , P (f ⊗ 1A ) = h ⊗ c1 + e ⊗ c2 + f ⊗ c3 , (4.26) with bi , ci ∈ B, i = 1, 2, 3. Using the structure equations above we see that there exist only two solutions: (A) a1 = 1, b2 = α with an invertible element α ∈ B, c3 = α −1 , and all other elements equal zero; and (B) a1 = −1, b3 = α with an invertible element α ∈B, c2 = α −1 , and all other elements equal zero. After an inner automorphism (given 0 1 by ) solution (B) is transferred to solution (A). Hence, we can assume that P −1 0 has the property P (h ⊗ 1A ) = h ⊗ 1B , P (e ⊗ 1A ) = e ⊗ α, P (f ⊗ 1A ) = f ⊗ α −1 ,
α ∈ B invertible.
(4.27)
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We can decompose P (sl(2) ⊗ A) into the weight spaces under the action of h ⊗ 1B = P (h ⊗ 1A ). The weight spaces h ⊗ B, e ⊗ B, and f ⊗ B correspond to the weights 0, 2, and −2 respectively. As [h ⊗ 1B , P (x ⊗ a)] = [P (h ⊗ 1A ), P (x ⊗ a)] = P ([h ⊗ 1A , x ⊗ a]) = P ([h, x] ⊗ a),
(4.28)
for a ∈ A, we obtain for x ∈ {h, e, f }, P (h ⊗ a) = h ⊗ Qh (a),
P (e ⊗ a) = e ⊗ Qe (a),
P (f ⊗ a) = f ⊗ Qf (a), (4.29)
with linear maps Qh , Qe , Qf : A → B. We will show that the map Qh is an algebra isomorphism. Let a1 , a2 ∈ A. From [e ⊗ a1 , f ⊗ a2 ] = [e, f ] ⊗ (a1 a2 ) = h ⊗ (a1 a2 ) one obtains after applying P that Qh (a1 a2 ) = Qe (a1 )Qf (a2 ). Next we consider [h ⊗ a, e ⊗ 1A ] = [h, e] ⊗ a = 2e ⊗ a. As α = Qe (1), we obtain after applying P that Qe (a) = Qh (a) · α. In the same way we get Qf (a) = Qh (a) · α −1 . Hence, Qh (a1 · a2 ) = Qh (a1 ) · Qh (a2 ) and Qh (1) = 1. This implies that Qh is indeed an isomorphism of associative algebras A → B. The proof shows that Proposition 4.9 is also true for the case that g is a reductive but non-abelian Lie algebra. In fact, it is only required that g admits a Lie subalgebra isomorphic to sl(2). Theorem 4.10. Let g be a finite-dimensional simple Lie algebra. Then the classical current algebra g = g⊗C[z−1 , z] is neither geometrically nor analytically rigid. Examples of nontrivial geometric deformations over the affine line C, resp. the algebra C[e], are s given by the current algebras G (e) of Krichever-Novikov type (4.18), (4.19) for every s ∈ C ∪ {∞}. s
Proof. Obviously for every s, G (e) defines a geometric deformation of g over C[e]. In s particular we get G (0) = g. As by Proposition 4.8 the function algebras are not isos morphic, Proposition 4.9 implies that the algebra G (e) is also not isomorphic to g as long as e = 0. Hence, restricted to every (algebraic-geometrically or even analytically) open neighbourhood of e = 0, the family will not be equivalent to the trivial family. By Definition 2.5 and 2.6, g is neither geometrically, nor analytically rigid. By a result of Lecomte and Roger [20] for g simple, the classical current algebra g is formally rigid. Hence Corollary 4.11. Despite their formal rigidity, the classical current algebras are neither geometrically nor analytically rigid. The results shown in this subsection extend to the centrally extended algebras. In particular, the classical affine Lie algebras g will be neither geometrically nor analytically rigid. Examples are given by the families of Theorem 4.6.
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5. Geometric Degenerations It might be quite instructive to identify geometrically all the algebras corresponding to the singular cubic situations. Above, by comparing the structure constants, we identified (s) G (0,0) and G for s = 1 or −2 with algebras which are related to the genus zero situation. The geometric scheme behind this is that in each case the desingularisation (or normalization) of the singular cubic is the projective line. By pulling back functions on the singular cubics we obtain functions on the desingularisation. But not necessarily all functions on the desingularisation will be obtained as pullbacks. One also has to keep track of the poles. In [30] the situation for meromorphic forms of arbitrary weights is analysed in complete detail. Three different situations appear. (I) All three e1 , e2 and e3 coincide. The normalization e1 + e2 + e3 = 0 implies necessarily that they are zero. We obtain the cuspidal cubic where the singular point is also a point of possible poles. Over the singular point there is only one point. Hence, we obtain an identification of the algebra A(0,0) with the full algebra C[z, z−1 ], and furthermore the identification of the current algebra G (0,0) with the classical current algebra g. (II) If two of the ei coincide (but not all three), we obtain the nodal cubic. Over the singular point we have two points on the desingularisation. Let the desingularisation be chosen such that the points α and −α lie above the singular point. We have to distinguish two sub-cases. (IIa) Either e1 = e2 = e3 or e1 = e3 = e2 , then the singular point (the node) will become a possible pole. This situation occurs if we approach points from ∗ . The algebra generated by the pullbacks will be the full function D1∗ ∪ D−2 algebra of the 3-point algebra V(α) . Hence, we see geometrically the already remarked isomorphy. On the level of the current algebra we get the identifi1 −2 cation of Y(α) with G (e) and G (e) . (IIb) If e1 = e2 = e3 , then the point of a possible pole will remain non-singular. ∗ This appears if we approach a point of D−1/2 . For the pullbacks of the functions it is now necessary that they have the same value at the points α and −α. Hence, all elements of the algebra generated by the pullbacks will have the same property. We describe this algebra in the following. Proposition 5.1. The set of elements zn , n even, An := n z − α 2 zn−2 = zn−2 (z2 − α 2 ), n odd,
(5.1)
for n ∈ Z form a basis of the subalgebra W(α) of meromorphic functions on P1 which are holomorphic outside 0 and ∞ and have the same value at α and −α. The algebra structure is given by for n or m even, An+m , An · Am = An+m − 2α 2 An+m−2 (5.2) +α 4 A for n, m odd. n+m−4 , For this subalgebra we have W(α) ∼ = A−1/2 and it is not isomorphic to C[z−1 , z].
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Proof. Obviously these elements lie in this subalgebra and form a basis of the subalgebra of sums of all even functions (without any restriction) and odd functions which vanish at ±α. Let f be a function fulfilling the conditions for being a member of W(α) . Decompose it into its symmetric and antisymmetric part, f = f1 + f2 ,
f1 (x) = 1/2(f (x) + f (−x)),
f2 (x) = 1/2(f (x) − f (−x)). (5.3)
Obviously f1 also fulfills the conditions, hence f2 too and we get f2 (−α) = f2 (α). Being an antisymmetric function this implies that f2 has to vanish at ±α, and f is a linear combination of the elements An . By setting α = i 3e21 we immediately see that ∼ A−1/2 . Hence by Proposition 4.8, W(α) is not isomorphic to C[z−1 , z]. W(α) =
Clearly again the algebras W(α) are isomorphic for different α = 0. The above mentioned identifications extend to the current algebras and we obtain another one-dimensional algebraic-geometric deformation family Z(α) of the current algebra. Proposition 5.2. In the two-parameter family (4.17), the current algebras for the singular cubic cases are isomorphic as follows: G (0,0) ∼ = g = g ⊗ C[z, z−1 ], 1 −2 G ∼ =G ∼ = g ⊗ V, −1/2 ∼ G = g ⊗ W.
(5.4)
Recall that here we denoted the family with the same letter as the isomorphy type of the generic member. These considerations extend to the centrally extended families of algebras introduced in Sect. 4.4, see Proposition A.1. 6. Cohomology Classes of the Deformations Let G t be a one-parameter deformation of the current algebra g with Lie structure [α, β]t = [α, β] + t k ω0 (α, β) + t k+1 ω1 (α, β) + · · · ,
(6.1)
such that ω = ω0 is non-vanishing. As explained in Sect. 2.1, the bilinear map ω will be an element of C2 (g, g). s For the global families G (e) over the affine line Ds appearing in Sect. 4 we obtain as the first nontrivial contribution the following two-cocycles (s = ∞): 0, for n or m even, ω(x ⊗ An , y ⊗ Am ) = (6.2) [x, y] ⊗ An+m−2 for n, m odd. For D∞ we get ω(x ⊗ An , y ⊗ Am ) =
0, [x, y] ⊗ An+m−4 ,
for n or m even, for n, m odd.
(6.3)
Here we used the symbols x ⊗ An for the basis elements x ⊗ zn also in the classical case. It should be pointed out that the families corresponding to the singular cubic situations are included for s = 1, −2 or s = −1/2.
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Proposition 6.1. Let g be an arbitrary finite-dimensional Lie algebra. The two-cocycles (6.2) and (6.3) for the algebra g are cohomologically trivial. Hence, considered as s infinitesimal deformations of g, the families G (e) are trivial. Proof. We verify directly that ω = d1 η for η : g → g defined by 0, n even, 0, n even, η(x ⊗ An ) = resp. (6.4) 1 1 − 2 x ⊗ An−2 , n odd, − 2 x ⊗ An−4 , n odd. Clearly, if g is a simple Lie algebra, the vanishing of the cohomology class follows from the formal rigidity of g ([20]). Remark. As an intermediate step we constructed (inside the category of commutative and associative algebras) nontrivial algebraic-geometric deformations As(e) of the algebra of Laurent polynomials C[z−1 , z]. One sees here the same effect as in the Lie case. The relevant cohomology theory is the Harrison cohomology (see Sect. 2.8). As C[z−1 , z] is the algebra of regular functions on the smooth affine curve P1 \ {0, ∞}, it is a smooth affine algebra and its Harrison two-cohomology vanishes [15, Thm.22]. Again we get that, despite the fact that this algebra is infinitesimally and formally rigid, there exist nontrivial local geometric families. See Kontsevich [17] for his concept of semi-formal deformations (related to filtrations of certain type) for possibly a way to overcome this discrepancy. Indeed, by the almost-gradedness of the involved families considered in this article, the families As(e) are semi-formal deformations in his sense. Further considerations in these directions have to be postponed to another article. As the Harrison two-cohomology vanishes, the defining cocycle for these families will be a coboundary. One verifies immediately that it is the coboundary of the linear form 0, n even, 0, n even, η(An ) = 1 resp. (6.5) 1 A , n odd, A , n odd. 2 n−2 2 n−4 A. Proof of Theorem 4.6 In this appendix we will show that the geometrically defined central extension (4.17) of the family G (e1 ,e2 ) has the form (4.23) if expressed for pairs of generators x ⊗ An , x ∈ g and An , n ∈ Z the (homogeneous) basis elements of A(e1 ,e2 ) . It is enough to show the
1 A dA expression (4.24) for 2πi n m for every pair of basis elements of the function CS algebra A(e1 ,e2 ) . As a side remark, this integral defines central extensions of the algebra of functions considered as abelian Lie algebras, see [32]. As the integration is over a separating cycle CS , the integral can be calculated by calculating residues at either one of the possible poles, (e1 , 0) or ∞. From the point ∞ the residue has to be taken with a minus sign. One possible way to calculate the residue is to change to the complex-analytic picture. This means we use the fact that X corresponds to the elliptic Weierstraß ℘-function and Y to its derivative ℘ . They are doubly-periodic functions on the complex plane with respect to the lattice = Z ⊕ τ Z with τ ∈ C, im τ > 0. We use the variable z for the complex variable in the plane. With this identification the complex one-dimensional torus C/ is analytically isomorphic to the (projective) elliptic curve. The variable τ fixes the isomorphy class of the torus, the
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parameters e1 , e2 , e3 the isomorphy class of the elliptic curve. Clearly, they are related. One relation which is important for us is 1 ℘ ( ) = e1 , 2
τ ℘ ( ) = e2 , 2
℘(
τ +1 ) = e3 . 2
(A.1)
Note also that the functions ℘ and ℘ depend on τ and hence also on the parameters e1 , e2 and e3 . We recall the following well-known facts from the theory of elliptic functions. The function ℘ fulfills the differential equation (℘ )2 = 4(℘ − e1 )(℘ − e2 )(℘ − e3 ) = 4℘ 3 − g2 ℘ − g3 .
(A.2)
We have e1 + e2 + e3 = 0,
g2 = −4(e1 e2 + e1 e3 + e2 e3 ),
g3 = 4(e1 e2 e3 ). (A.3)
The function ℘ is an even meromorphic function with poles of order two at the points of the lattice and holomorphic elsewhere. The function ℘ is an odd meromorphic function with poles of order three at the points of the lattice and holomorphic elsewhere. It has zeros of order one at the points 1/2, τ/2 and (1 + τ )/2 and all its translates under the lattice. The zeros are of order one. For the Laurent series expansion at z = 0 we obtain ℘ (z) =
1 g2 (1 + z4 + O(z6 )), z2 20
℘ (z) − e1 =
1 g2 (1 − e1 z2 + z4 + O(z6 )). z2 20 (A.4)
In terms of these functions the basis elements An can be expressed as A2k = (℘ − e1 )k ,
A2k+1 =
1 ℘ · (℘ − e1 )k−1 2
k ∈ Z.
(A.5)
Note that (℘ − e1 ) has a pole of order two at z = 0 and a zero of order two at z = 1/2. It has a Laurent expansion in even powers of z. Hence the same is true for A2k . Further1 d more, A2k+1 (z) = 2k dz A2k (z). This allows us to determine the Laurent expansion of A2k+1 easily. Its Laurent expansion consists of only odd powers of z. Our first conclusion is that the differential A2k+1 A2l dz has only even terms in its Laurent expansion. Hence, as the residue is the coefficient of 1z , there is no residue, and the cocycle values evaluated for pairs of basis elements of different parity are zero, as claimed (4.24). Due to the residue theorem (on the torus) the total residue of any differential has to vanish. Hence, there will be only a residue at one point if the differential has poles at both points, z = 0 and z = 1/2. In the even case, A2k A2l dz, the pole order at 0 is 2(k + l) + 1, and at 1/2 it is (−2k) + (−2l + 1) = −2(k + l) + 1. Hence, a possibly non-vanishing value could appear only if l = −k (or, equivalently m = −n). In the odd case, A2k+1 A2l+1 dz, we have poles at 0 of order (2k + 1) + (2l + 2) = 2(k + l) + 3, and at 1/2 of order (−2k + 1) + (−2l + 2) = −2(k + l) + 3. Hence, there could only be a contribution if l = −k − 1, −k, or −k + 1 corresponding to m = −n, −n + 2, or −n + 4.
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To calculate the residue we have to consider the Laurent expansion of the generators and their derivatives. Starting from (A.4) we get k − 1 2 g2 4 1 2 6 e + z + O(z ) , (A.6) A2k = 2k 1 − ke1 z + k z 2 1 20 1 A2k = 2k+1 (−2k + O(z2 )), z 1 1 A2k+1 = A2k = 2k+1 − 1 + (k − 1)e1 z2 2k z k − 1 2 g2 4 6 e + z + O(z ), +(−k + 2) 2 1 20 1 A2k+1 = 2k+2 2k + 1 + (k − 1)(−2k + 1)e1 z2 z k − 1 2 g2 4 6 +(−k + 2)(−2k + 3) e + z + O(z ) , (A.7) 2 1 20 For the even pairing, i.e. n = 2k, m = 2l, we obtain in case l = −k, 1 (2k + O(z2 )). (A.8) z Hence
the residue at z = 0 is 2k = m, and as it has to be taken negatively, we obtain 1 2πi CS An dA−n = −n, as claimed in (4.24). For the odd pairing, i.e. n = 2k + 1, m = 2l + 1, we have to multiply the Laurent expansions of A2k+1 and A2l+1 . The resulting Laurent series is 1 2 4 6 a(k, l) + b(k, l)z + c(k, l)z + O(z ) dz, (A.9) z2(k+l)+3 with a(k, l), b(k, l) and c(k, l) obtained by collecting the corresponding contributions to these orders. In particular, we obtain A2k A−2k dz =
a(k, l) = −(2l + 1),
b(k, l) = e1 ((k − 1)(2l + 1) + (l − 1)(2l − 1)). (A.10)
Now the residue at z = 0 can be easily calculated: For l = −k − 1 we obtain as residue a(k, −k − 1) = (2k + 1) = n. For l = −k the residue is b(k, −k) = −3e1 (−n + 1). For l = −k + 1 the residue is c(k, −k + 1) = −(e1 − e2 )(2e1 + e2 )(−n + 2).
(A.11)
For the last calculation we have to express g2 by the ei ’s as given in (A.3) and replace e3 = −(e1 + e2 ). Finally, it should not be forgotten to change the sign to obtain the cocycle values claimed in (4.24). Strictly speaking, this derivation via the elliptic functions is valid above the points outside of the collection of lines D1 ∪ D−2 ∪ D−1/2 . But being a cocycle is a closed condition, hence the bilinear form will also be a cocycle for the algebras not corresponding to elliptic algebras. It will be obtained by specializing the values of e1 and e2 . Nevertheless, it is an easy exercise to calculate the cocycle for the genus zero algebras using again residues. In this case the functions are rational functions, hence the calculations will be easier. For a convenient reference, let us note here the form of the cocycle for the singular situation.
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Proposition A.1. (a) For the current algebra G (0,0) = g the centrally extended algebra is given by the cocycle (p ∈ C) −n γ (x ⊗ An , y ⊗ Am ) = p · β(x, y) · (−n)δm .
(A.12)
(b) For the three-point genus zero family of current algebras Y(α) (4.22) the family of centrally extended algebras is given by the cocycle 1 γ (x ⊗ An , y ⊗ Am ) = p(α) · β(x, y) · An dAm , (A.13) 2π i CS with a polynomial p in α and −n , n, m even, −nδm 1 An dAm = 0, n, m different parity, (A.14) 2π i CS −nδ −n + α 2 (−n + 1)δ −n+2 , n, m odd. m m (c) For the family of current algebras Z(α) defined in Sect. 5 the family of centrally extended algebras is given by the cocycle 1 γ (x ⊗ An , y ⊗ Am ) = p(α) · β(x, y) · An dAm , (A.15) 2π i CS with a polynomial p in α and −n , −nδm n, m even, 1 0, n, m different parity, (A.16) An dAm = −n − 2α 2 (−n + 1)δ −n+2 −nδ 2π i CS m m +α 4 (−n + 2)δ −n+4 , n, m odd. m B. A Different Proof of Theorem 4.6
The bilinear form γ (f, g) = 2π1 i CS f dg on the function algebra A = A(e1 ,e2 ) appearing in the definition of the cocycle for the current algebra is a cocycle for A (considered as an abelian Lie algebra) defining a central extension of it, i.e. an infinite dimensional Heisenberg algebra. In [32] it was shown that this cocycle is up to multiplication with a constant the unique local and multiplicative cocycle for A. Note that because A is abelian, there are no non-trivial coboundaries. A cocycle is called local if there are constants M and N such that γ (An , Am ) = 0,
implies
M ≤ n + m ≤ N.
(B.1)
Being a local cocycle says that the central extension defined via the cocycle is almostgraded. A cocycle is called multiplicative if γ (fg, h) + γ (gh, f ) + γ (hf, g) = 0,
f, g, h ∈ A.
(B.2)
In the quoted article the uniqueness was shown by applying a recursive procedure using the locality and the multiplicativity of the cocycle and the almost-graded structure of the 1 algebra A. From this it is clear that the total cocycle 2πi CS f dg is already fixed by one non-vanishing cocycle value (let’s say γ (A2 , A−2 )) and the structure constants (4.15)
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of the algebra. Instead going through the Laurent series calculations above, one could equally well go through the combinatorics of the recursion in [32] using the structure of A. We will supply such an alternative proof of (4.24) in this appendix. From this proof the similarity of the structure equations (4.15) with the formulas for the central extensions (4.24) will become clear.
1 First note that as an antisymmetric and bilinear form, γ (f, g) = 2πi CS f dg defines a two-cocycle for the abelian Lie algebra A. Furthermore, note that there does not exist any non-trivial coboundary. Recall that we are working over a compact Riemann surface, resp. a smooth projective curve over C. By the residue theorem the total residue of every differential has to vanish. Hence, a differential having a pole with a (point-)residue has to have at least another pole. This implies that with respect to the grading introduced, the cocycle is local (see (B.1)). This can be seen by estimating the pole orders of the differential An dAm at the points 0 and 1/2. As the (point-)residue of an exact differential of a meromorphic function is zero, we obtain 1 1 1 1 0= d(f gh) = f gd(h) + ghd(f ) + hf d(g). 2π i CS 2πi CS 2πi CS 2π i CS (B.3) This condition is exactly the multiplicativity of the cocycle γ . Recall the structure of the algebra: An+m , n or m even, An · Am = (B.4) An+m + aAn+m−2 + bAn+m−4 , n and m odd, with a = 3e1 and b = (e1 − e2 )(2e1 + e2 ). For the set of values γ (An , Am ) we introduce its level l = n + m. In the following we will use induction on the level l. 1. If l < 0 then γ (An , Am ) = γ (An , A−n+l ) = 0. This follows immediately from the fact that γ is a local cocycle (see [32], where the inverted grading was used). Alternatively, it can be easily checked by showing that there will be no pole at z = 0 as long as l < 0. 2. We have to calculate one cocycle value for normalization. −1 d γ (A2 , A−2 ) = −γ (A−2 , A2 ) = resz=0 (℘ (z) − e1 ) (℘ (z) − e1 ) dz = resz=0 (℘ (z) − e1 )−1 ℘ = −2. (B.5) 3. Lemma B.1. γ (An , A0 ) = 0 for all n ∈ Z. Proof. γ (An , A0 ) = γ (An , A0 · A0 ) = −γ (An · A0 , A0 ) − γ (A0 · A0 , An ) = −γ (An , A0 ) + γ (An , A0 ) = 0. (B.6) Above we used the multiplicativity. Again with the multiplicativity we obtain
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Lemma B.2. γ (f · f, f ) = 0. 4. Let l = 0, and n be of arbitrary parity. We calculate γ (An+1 , A−(n+1) ) = γ (An A1 , A−(n+1) ) = −γ (A1 A−(n+1) , An ) − γ (A−(n+1) An , A1 ) = −γ (A−n , An ) − γ (A−1 , A1 ) = γ (An , A−n ) + γ (A1 , A−1 ).
(B.7)
Here certain remarks are necessary, as this chain of equalities is not that innocent as it looks. For the first equality note that if n is even we can indeed replace An+1 by An A1 , using (B.4). If n is odd then An+1 = An A1 − aAn−1 − bAn−3 . But we have γ (An−1 , A−(n+1) ) = γ (An−3 , A−(n+1) ) = 0, as their levels are −2, resp. −4, and for those levels the cocycle is vanishing. Hence, indeed the first inequality is true for every parity. Using the multiplicativity, we replace the one cocycle by the negative of its two partner cocycles. Now we use the same kind of arguments as above to replace the products for them. Altogether we obtain a simple recursion relation which has as a unique solution γ (An , A−n ) = n · γ (A1 , A−1 ) = (−n).
(B.8)
The last equality follows from the cocycle value we calculated in Step 2. 5. We consider (n, −n + l) with l = 0, such that either both entries are even (this says l is even), or n is odd and the other entry is even (this says l is odd). We claim that all cocycle values for such pairs vanish. For l even, we will show the claim directly. For l odd we will use induction. First note that for l negative the values will be zero by locality. Hence the start of the induction is in the odd case trivially true. We calculate γ (An , A−n+l ) = γ (An−2 A2 , A−n+l ) = −γ (A2 A−n+l , An−2 )−γ (A−n+l An−2 , A2 ) = γ (An−2 , A−(n−2)+l )+γ (A2 , Al−2 ). (B.9) Lemma B.3. For l = 0 we have γ (A2 , Al−2 ) = 0. Proof. By the locality γ (A2 , Al−2 ) = 0 if l ≤ 0. First we have to consider l = 1, 2, 3, 4 individually: For l = 1 we get γ (A2 , A−1 ) = γ (A1 A1 , A−1 ) = −γ (A1 A−1 , A1 ) − γ (A−1 A1 , A1 ) = −2γ (A−1 A1 , A1 ) = −2(γ (A0 , A1 ) + aγ (A−2 , A1 ) + bγ (A−4 , A1 )) = 0. (B.10) Here we used that the cocycle vanishes if the level is negative, the multiplicativity and Lemma B.1. For l = 2 we have γ (A2 , A0 ) = 0 by Lemma B.1. For l = 3 we calculate γ (A2 , A1 ) = γ (A1 A1 − aA0 − bA−2 , A1 ) = γ (A1 A1 , A1 ) = 0. (We used also Lemma B.2.) For l = 4 we have γ (A2 , A2 ) = 0 by the antisymmetry.
(B.11)
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Now let l ≥ 5. We showed above that γ (Ar , A−r+l ) − γ (Ar−2 , A−(r−2)+l ) + γ (A2 , Al−2 ) = 0.
(B.12)
If l is even we let r = 4, 6, . . . , l − 2 and sum this (l − 4)/2 equations up. As a sum we obtain γ (Al−2 , A2 ) − γ (A2 , Al−2 ) −
l−4 γ (A2 , Al−2 ) = 0. 2
(B.13)
Hence, l−4 γ (A2 , Al−2 ) = 0, 2
(B.14)
and as l > 4, γ (A2 , Al−2 ) = 0. For l odd we sum up the equations for r = 3, 5, . . . , l − 2. These are (l − 3)/2 equations. The sum calculates to γ (A1 , Al−1 ) +
l−1 γ (A2 , Al−2 ) = 0. 2
(B.15)
We calculate γ (A2 , Al−2 ) = γ (A1 A1 , Al−2 ) − aγ (A0 , Al−2 ) − bγ (A−2 , Al−2 ) = γ (A1 A1 , Al−2 ),
(B.16)
as the other summands vanish by induction. Hence, γ (A2 , Al−2 ) = −γ (A1 Al−2 , A1 ) − γ (Al−2 A1 , A1 ) = 2γ (A1 , Al−1 + aAl−3 + bAl−5 ).
(B.17)
By induction what remains is γ (A2 , Al−2 ) = 2γ (A1 , Al−1 ). Hence, from (B.15) it follows that also in this case γ (A2 , Al−2 ) = 0.
(B.18)
This proof shows also Lemma B.4. For l odd we have γ (A1 , Al−1 ) = 0. If n is even from (B.9) it follows that γ (An , A−n+l ) = n2 γ (A2 , Al−2 ). Hence by Lemma B.3, all cocycle values will vanish. If n is odd, l is odd and we get by Lemma B.4 and Lemma B.3 that the starting value and the increment for the recursion are zero. Hence, also here all cocycle values will vanish. 6. It remains to consider the case that both entries in (n, −n + l) are odd, and l > 0. In any case, l will be an even number, γ (An , A−n+l ) = γ (An−1 A1 , A−n+l ) = −γ (A1 A−n+l , An−1 ) − γ (A−n+l An−1 , A1 ) = −γ (A−n+l+1 + aA−n+l−1 + bA−n+l−3 , An−1 ) − γ (Al−1 , A1 ). (B.19)
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In the first expression there are only (even, even) combinations, which have only a non-vanishing value if they are (k, −k). Altogether we obtain γ (An , A−n+l ) = −a(n − 1)δl2 − b(n − 1)δl4 + γ (A1 , Al−1 ).
(B.20)
For l = 2 we obtain γ (A1 , Al−1 ) = γ (A1 , A1 ) = 0. Hence, γ (An , A−n+2 ) = a · (−n + 1). For l = 4 we calculate γ (A1 , A3 ) = γ (A1 , A2 A1 ) = −γ (A1 , A1 A2 ) − γ (A2 , A1 A1 ).
(B.21)
This implies (using the normalization calculated in the second step) 1 γ (A1 A1 , A2 ) 2 1 b = γ (A2 + aA0 + bA−2 , A2 ) = γ (A−2 , A2 ) = b. 2 2
γ (A1 , A3 ) =
(B.22)
With (B.20) we obtain γ (An , A−n+4 ) = b · (−n + 2). For l > 4 we have γ (An , A−n+l ) = γ (A1 , Al−1 ). This says that for level l all values are equal to the same constant γ (A1 , Al−1 ). This is only possible if this constant is zero, as γ (A1 , Al−1 ) = −γ (Al−1 , A1 ) = −γ (Al−1 , A−(l−1)+l ) = −γ (A1 , Al−1 ). (B.23) Hence, γ (An , A−n+l ) = 0 for l > 4. This closes the calculation of the cocycle values (4.24). Clearly, this calculation also shows (A.14) and (A.16). Remark. In fact, we even showed more. We showed that for each commutative and associative algebra A with structure (B.4) and the obvious almost-grading every local cocycle for its abelian Lie structure which is multiplicative (with respect to the associative structure) is given as a scalar times the cocycle (4.23), of course instead of 3e1 we get a and instead of (e1 − e2 )(2e1 + e2 ) we get b. Acknowledgement. Both authors thank different institutions for hospitality experienced during the preparation of this article. A. F. and M. Sch. thank the Erwin-Schr¨odinger Institute (ESI) in Vienna, M. Schl. ´ the E¨otv¨os Lor´and University in Budapest, and the Institut des Hautes Etudes Scientifiques (IHES) in Bures-sur-Yvette. Discussions with M. Kontsevich, P. Michor and E. Vinberg are gratefully acknowledged. The work was partially supported by grants OTKA T034641 and T043034. Furthermore, we thank the referee for suggestions which helped to improve the presentation.
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35. Schlichenmaier, M., Sheinman, O.K.: Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras. Russ. Math. Surv. (Usp. Math. Nauk.) 54, 213– 250 (1999) 36. Schlichenmaier, M., Sheinman, O.K.: Knizhnik-Zamolodchikov equations for positive genus and Krichever-Novikov algebras. Russian Math. Surv. 59(4), 737–770 (2004) 37. Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. Adv. Stud. Pure Math. 19, 459–566 (1989) Communicated by L. Takhtajan
Commun. Math. Phys. 260, 613–640 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1421-7
Communications in
Mathematical Physics
Products of Floer Cohomology of Torus Fibers in Toric Fano Manifolds Cheol-Hyun Cho Department of Mathematics, Northwestern University, Evanston, IL 60208, USA. E-mail: [email protected] Received: 6 December 2004 / Accepted: 8 March 2005 Published online: 16 August 2005 – © Springer-Verlag 2005
Abstract: We compute the ring structure of Floer cohomology groups of Lagrangian torus fibers in some toric Fano manifolds continuing the study of [CO]. Related A∞ -formulas hold for a transversal choice of chains. Two different computations are provided: a direct calculation using the classification of holomorphic discs by Oh and the author in [CO], and another method by using an analogue of divisor equation in Gromov-Witten invariants to the case of discs. Floer cohomology rings are shown to be isomorphic to Clifford algebras, whose quadratic forms are given by the Hessians of functions W , which turn out to be the superpotentials of Landau-Ginzburg mirrors. In the case of CP n and CP 1 × CP 1 , this proves the prediction made by Hori, Kapustin and Li by B-model calculations via physical arguments. The latter method also provides correspondence between higher derivatives of the superpotential of LG mirror with the higher products of the A∞ (or L∞ )-algebra of the Lagrangian submanifold.
1. Introduction Floer theory of Lagrangian intersections has been proved to be a powerful technique in symplectic geometry. Also since the “homological mirror symmetry” conjecture by Kontsevich [K], it has become a much more exciting field of mathematics, which yet has a long way to be fully understood. Recently, Fukaya, Oh, Ohta and Ono constructed an A∞ -algebra of a Lagrangian submanifold and Floer homology in a general setting in their beautiful work [FOOO]. But the construction is highly non-trivial to overcome several technical problems. The first problem is the well-definedness of the moduli space of J -holomorphic discs compatible for all homotopy classes. It was observed in [FOOO], that standard Kuranishi perturbation does not produce compatible and transversal moduli space in general. Another problem is that even if moduli spaces of J -holomorphic discs are well-defined, it does not directly produce an A∞ -algebra since one has to work at the chain level.
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In [CO], Yong-Geun Oh and the author have explicitly described the moduli space of holomorphic discs in the case of Lagrangian torus fibers in toric Fano manifolds, and used that information to compute Floer cohomology groups. A combinatorial description of a fiber whose Floer cohomology is non-vanishing was found, and for such a fiber, the Floer cohomology was in fact isomorphic to singular cohomology as a module. It was shown that all holomorphic discs in these cases are transversal. To compactify the moduli space, we need an additional assumption regarding the behavior of holomorphic spheres on a toric Fano manifold (see Assumption 3.1). In this paper, we first consider a related A∞ -algebra which is defined transversally. Namely, fiber products with various chains in the Lagrangian submanifold L in the definition of an A∞ -algebra can be made transversal for the generic choice of chains. This gives a partial A∞ -algebra, but products on the cohomology of these A∞ -algebras are shown to be well-defined. How to obtain an actual A∞ -algebra from this partial algebra is an interesting question. With skewsymmetrization in this toric Fano case, these partial A∞ -algebras gives well-defined L∞ -algebras. On the other hand, recently Fukaya has constructed an A∞ -algebra on DeRham complex of Lagrangian submanifolds. A computation in toric Fano case can be carried out in the DeRham setting, which will produce actual A∞ -algebra. Then we show that Floer cohomology ring H F BM (L; J0 ) is isomorphic to a Clifford algebra Cl(V , Q) where Q is a symmetric bilinear form. It is very interesting that the symmetric bilinear form Q we obtained exactly agrees with the Hessian of the superpotential W of the mirror Landau-Ginzburg model studied by Hori and Vafa [HV]. (This is related to homological mirror symmetry conjecture between A-model in Fano manifolds and B-model in Landau-Ginzburg mirror.) In particular, the Floer cohomology of the Clifford torus T n in CP n is isomorphic to the Clifford algebra with n generators as a ring. Such product structures in the Clifford torus T n in CP n and T 1 × T 1 in CP 1 × CP 1 have been conjectured by Hori and Kapustin and Li [KL], recently in general by [KL2] from the calculation on B-model side using physical arguments. Mathematical account of the product structure on B-model side looks plausible considering the paper by Orlov [O]. We provide two ways of computing the product structure. First, we provide direct computations exploiting the classification of all holomorphic discs with boundary on L by Oh and the author ([CO]). Another method is by using an analogue of divisor equation for discs, which is introduced in section 6. The latter method easily provides the general correspondence between higher derivatives of the superpotential of LG mirror with the higher products of A∞ (or L∞ )-algebra of Lagrangian submanifold. This extends the correspondence proved by Oh and the author in [CO] that obstruction cochain m0 = l0 agrees with the superpotential itself and non-vanishing of Floer cohomology corresponds to the critical points of the superpotential W . These l∞ -products are invariant under the perturbation of an almost complex structure. We also provide an explicit filtered chain map between singular cochain complex and Bott-Morse Floer complex in the case of torus fibers L in toric Fano manifolds, which induces an isomorphism in cohomology in case Floer homology is non-vanishing.
2. A∞ -Algebra of Lagrangian Submanifold In this section we recall the construction of the A∞ -algebra of a Lagrangian submanifold. In fact, we will provide a transversal version (partial A∞ -algebra) which is suitable for our purposes. (This version is only suitable for the case when the moduli space is already well-defined.)
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The A∞ -algebra in this case naturally arises from the stable map compactification of the moduli spaces of holomorphic discs. The moduli space of a disc with n + 1 boundary marked points, Mn+1 , can be seen also as a compactification of a configuration space of n − 2 points on an interval [0, 1]. (By Aut (D 2 ), send n + 1, 0, 1st marked points to 1, ∞, 0 where we identify D 2 with the upper-half plane.) The latter gives the well-known Stasheff Polytope [S1]. We first recall the definition of the (non-unital) A∞ -algebra introduced by Stasheff [S1]. Let A = ⊕i∈Z Ai be a Z-graded module over R, where R is a commutative ring with unit. As usual, we denote its suspension by A[1]i = Ai+1 . Definition 2.1. A structure of the (non-unital) A∞ -algebra on A is given by a series of R-module homomorphisms mk : A⊗n → A[2 − n] for non negative integer k, satisfying quadratic equations (−1)deg x1 +···+deg xi−1 +i−1 (2.1) k1 +k2 =k+1 i
mk1 (x1 , . . . , mk2 (xi , . . . , xi+k2 −1 ), . . . , xk ) = 0. In the transversal version, the above formula will only hold on a dense transversal sequence of chains for each k. Now we recall the setting for the objects of the chain complex. We refer readers to [FOOO] Appendix A for a complete explanation about introducing this setup. Let C ∗ (L; nov ) be the set of currents on L realized by geometric chains as follows: For a given (n-k)-dimensional geometric chain [P , f ], we consider the current T ([P , f ]) which is defined as follows: The current T ([P , f ]) is an element in D k (M; R), where D k (M; R) is the set of distribution valued k-forms on M : For any smooth (n-k)-form ω, we put T ([P , f ]) ∧ ω = f ∗ ω. (2.2) M
P
This defines a homomorphism T : Sn−k (M; Q) → D (M; R), k
where Sn−k (M; Q) is the set of all (n-k) dimensional geometric chains with Q-coefficient. k Let S (M, Q) be the image of the homomorphism T . We extend the coefficient ring Q to nov . Then we set k
C k (L; nov ) := S (M, nov ).
(2.3)
Since we consider the elements in the image of T , if the image of the map f of the geometric chain [P , f ] is smaller than the expected dimension, then it gives 0 as a current. This fact will be used crucially later on. Also, note that the map T is not injective, hence some elements get identified under the map T . Also note that we take the whole image of T (instead of taking a countable subset of it) as transversality of fiber products in the definition of mk is achieved by choosing generic chains. The classical part of the maps {mk } are defined as follows, which is different from that of [FOOO] (In [FOOO], mk,0 defines an A∞ -algebra of singular cochains.)
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Definition 2.2. The maps mk,0 for k = 0, 1, . . . on C ∗ (L; nov ) are transversally defined by the following maps. For [P , f ], [Q, g] ∈ C ∗ (L; Q), (1) m0,0 = 0. (2) m1,0 ([P , f ]) = (−1)n [∂P , f ]. (3) m2,0 ([P , f ], [Q, g]) = (−1)degP (degQ+1) [f (P ) ∩ g(Q), i] = 0, where i is an embedding into L. (4) for k ≥ 3, mk,0 ≡ 0.
(2.4)
We extend the above maps linearly over nov . The notation ∂ here is the usual boundary operator for singular homology. Now, the quantum contribution part is defined in the same way as in [FOOO]. Definition 2.3 [FOOO]. (1) For a geometric chain [P , f ] ∈ C g (L : Q) and non-zero β, define m0,β = [M1 (β), ev0 ],
(2.5)
m1,β [P , f ] = [M2 (β) ev1 ×f P , ev0 ].
(2.6)
(2) For each k ≥ 2, non-zero β, for geometric chains [P1 , f1 ] ∈ C g1 (L : Q), . . . , [Pk , fk ] ∈ C gk (L : Q) (i.e. dimension of [Pi , fi ] as a chain is n − gi ), define mk,β ([P1 , f1 ], . . . , [Pk , fk ]) = (−1) [Mmain k+1 (β) (ev1 ,...,evk ) ×(f1 ,...,f2 ) (P1 × · · · × Pk ), ev0 ].
(2.7)
Here is a sign assigned as follows: = (n + 1)
j k−1
deg(Pi ).
(2.8)
j =1 i=1
(3) Then we define the maps mk (k ≥ 0) by mk ([P1 , f1 ], . . . , [Pk , fk ]) =
mk,β ([P1 , f1 ], . . . , [Pk , fk ])
β∈π2 (M,L)
⊗T Area(β) q µ(β)/2 . Remark 2.4. Here Mk (β) is a compactified moduli space of J -holomorphic discs with k marked point on ∂D 2 . Recall that Mk (β) for k ≥ 3 has several connected component. By the ordering of the k marked points on ∂D 2 and by Mmain (β) we denote the connected k component where marked points z1 , . . . , zk lie cyclically on ∂D 2 counter-clockwise. Also, the fiber products defined above are not always transversal, and we discuss this issue in Sect. 3. Here we recall the dimension formula of mk,β when the involved fiber product is transversal.
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Proposition 2.1 ([FOOO] Proposition 13.16). For non-zero β, when transversal, mk,β ((P1 , f1 ), . . . , (Pk , fk )) ∈ Cn−k
i=1 gi +µ(β)−2+k
(L : Q).
Proposition 2.2 (cf. [FOOO]). These {mk } maps satisfy the A∞ formulas (2.1) for transversal sequence of chains in C ∗ (L; nov ). Proof. This is essentially the theorem proved in [FOOO]. We recall its proof for the convenience of readers and explain the changes made for mk,0 . For simplicity, we recall the proof only for the third A∞ -formula. Consider the moduli space of J -holomorpic discs intersecting chains P and Q (see Fig. 1), (β)ev1 ,ev2 ×f,g (P × Q), ev0 ). m2,β (P , Q) = (Mmain 3
(2.9)
Now, we consider all possible stable map compactification of this moduli space and its image under the evaluation map. The limit configurations of codimension 1 of the image can be written as follows. See Fig. 1, where each figure corresponds to the following terms: m2,β (P , Q) → m2,β2 (m1,β1 (P ), Q), m2,β2 (P , (m1,β1 (Q), ),
(2.10)
m3,β2 (P , Q, m0,β1 ), m3,β2 (P , m0,β1 , Q), m3,β2 (m0,β1 , P , Q), m1,β1 (m2,β2 (P , Q)). (2.11) Degenerations into several (three or more) disc components or sphere bubbles also occur. But if transversalities are satisfied for such singular strata with positivity assumptions on a Lagrangain submanifold, such strata should be of codimension 2 or more, hence they do not contribute to the A∞ formulas. Now, these limit configurations can be written into an A∞ -formula up to sign: ∂(m2,β (P , Q)) = ±m2,β2 (m1,β1 (P ), Q) ± m2,β2 (P , (m1,β1 (Q), ) ±m3,β2 (P , Q, m0,β1 ) ± m3,β2 (P , m0,β1 , Q) ± m3,β2 (m0,β1 , P , Q) ±m1,β1 (m2,β2 (P , Q)).
Fig. 1. Limit configurations of (2.9) of codimension 1
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This is the third A∞ -formula in (2.1) up to sign, and other formulas can be obtained in a similar fashion by choosing a mk,β (P1 , . . . , Pk ) for general k in (2.9). Now we justify the changes made in the definitions of mk,0 ≡ 0 for k ≥ 3. Consider one of the m3,0 term appeared in the above configuration when β2 = 0, or generally one may consider the geometric chain m3,0 (P , Q, R). The dimension of the image under the evaluation map of m3,0 (P , Q, R) is always smaller than the virtual dimension of the moduli space: The reason is that the evaluation map of a constant disc forgets the moduli parameter. Namely, before evaluation, there is a parameter describing the position of four marked points on a disc. Recall that the moduli space of 4 marked points on ∂D 2 up to automorphisms of D 2 is diffeomorphic to R (see [FOh]). But as we evaluate on a constant disc, the image is always a point, while the moduli parameter is lost under the evaluation map. Hence, such a term m3,0 (P , Q, R) is of codimension 1 by the virtual dimension, but its actual image is of codimension 2. Hence terms involving m3,0 do not appear in the A∞ formula, which is obtained by considering the codimension 1 boundary of the image of the chain (2.9) under evaluation map. This phenomenon always happens for mk,0 for any k ≥ 3 because of the same reason. Hence we may set (transversally) mk,0 ≡ 0 for k ≥ 3. Note that in [FOOO], the evaluation maps of constant homotopy class are also perturbed by moduli parameters, so that the image has the same dimension as virtual dimension unlike our setting. Also note that m2,0 , m1,0 does not vanish as there are no moduli parameters in these cases. This proves the proposition.
Now, because of the presence of m0 terms, m21 = 0 does not always hold. Hence, Floer homology groups are not well-defined in general. Obstructions for the well-definedness of Floer cohomology was studied in [FOOO]. In an unobstructed case, one can deform the chain complex in a suitable way so that m21 = 0 holds. For the case of torus fibers in a toric Fano manifold, it is (weakly) obstructed, in which case Floer cohomology itself is well-defined. A related phenomenon in the language of A∞ -algebra is that m0 terms disappear from the A∞ -formula. Proposition 2.3 (compare [FOOO] Proposition 7.1). Let xi be an element in C ∗ (L; nov ) for i = 0, . . . , k, for a Lagrangian torus fiber L in toric Fano manifolds. Then, when transversal, we have mk+1 (x1 , . . . , [L], . . . , xk ) = 0, k ≥ 2, k = 0, m2 ([L], x0 ) = (−1)deg(x0 ) m2 (x0 , [L]) = x0 . Namely, [L] behaves as a strict unit. Proof. This was proved in [FOOO], except that we do not need to use a homotopy unit argument. Recall that the forget maps commute with the evaluation maps obviously in our case, whereas they do not commute in [FOOO] because of the perturbation of evaluation maps at marked points. We give the proof of the proposition here for the convenience of readers. Note that the proposition holds by the definition of mk+1,0 for k ≥ 2. Hence it is enough to show that mk,β (x1 , . . . , [L], . . . , xk ) = 0 for k ≥ 2 with non-zero β ∈ π2 (M, L), and the statement about m2,0 .
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Note that the condition for the image of a marked point to meet the fundamental chain [L] is redundant since it always meets L. Hence, ev0 Mk+2 (β) ev × (x1 × · · · × [L] × · · · × xk ) ⊂ ev0 (Mk+1 (β) ev × (x1 × · · · × xk )).
(2.12)
The dimension of RHS is n−
deg(xi ) + µ(β) − 2 + k,
where as the virtual (expected) dimension of LHS is n−
deg(xi ) + µ(β) − 2 + k + 1.
Hence, the actual image has smaller dimension than the expected dimension, which becomes zero in the language of currents. The case of m2,0 follows from the sign convention of [FOOO].
The second formula implies that the Floer cohomology is well-defined in this case as observed in [CO] and [C] Proposition 3.18; in this case it was shown that m0 (1) = N ei i=1 [L] ⊗ T q is a multiple of the fundamental chain. Here, we write the first three A∞ -formulas where m0 terms are dropped because of the above proposition. 0 = m1 ◦ m1 , 0 = m2 (m1 (x), y) + (−1)deg(x)+1 m2 (x, m1 (y)) + m1 (m2 (x, y)),
(2.13) (2.14)
0 = m1 (m3 (x, y, z)) + m2 (m2 (x, y), z) + (−1)deg(x)+1 m2 (x, m2 (y, z)) (2.15) +m3 (m1 (x), y, z) + m3 (x, m1 (y), z) + m3 (x, y, m1 (z)). The first equation implies that m1 defines the cochain complex. The second equation implies that m2 defines a product of the cohomology up to sign. For x, y, z ∈ H F BM (L; J0 ), we have m1 (x) = m1 (y) = m1 (z) = 0. Therefore the third equation implies the associativity of the product up to sign, m2 (m2 (x, y), z) + (−1)deg(x)+1 m2 (x, m2 (y, z)) = 0.
(2.16)
To define an associative product (with correct sign) on cohomology, one should make the following change of signs. Definition 2.5. We define m 1 (P ) = (−1)degP m1 (P ), m 2 (P , Q) = (−1)degP (degQ+1) m2 (P , Q).
(2.17) (2.18)
Remark 2.6. The first sign appears due to a acohomological sign convention. The second sign appears due to the sign convention of [FOOO].
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The resulting A∞ -formulas for the new { mk } are m2 (x, y)) = m 2 ( m1 (x), y) + (−1)deg x m 2 (x, m 1 (y)), m 1 ( m 2 ( m2 (x, y), z) = m 2 (x, m 2 (y, z)) 2 defines an graded associative product on H F BM for x, y, z ∈ H F BM (L; J0 ). Hence m (L; J0 ). For example, with the new sign, the classical cup product part of m 2 can be written as m 2,0 (P1 , P2 ) = P1 ∩ P2 . Also associativity in the classical level is just (P1 ∩ P2 ) ∩ P3 = P1 ∩ (P2 ∩ P3 ). 3. Transversality In this section, we discuss the issues regarding the moduli space of J -holomorphic discs and the transversality of A∞ -algebra. 3.1. Moduli spaces. We first recall the following theorem. Theorem 3.1 ([CO]). Holomorphic discs in toric manifolds with boundary on any Lagrangian torus fiber are Fredholm regular, i.e., its linearization map is surjective. Hence the moduli space of holomorphic discs (before compactification) is a manifold of the expected dimensions. As we try to compactify the moduli space, we may have strata with sphere bubbles. In general toric Fano manifolds, it is already known that holomorphic spheres are not always Fredholm regular. Hence in the compactification of holomorphic discs, some strata (with sphere bubble) may not have the expected dimension. But since we only evaluate at the boundary of the discs (not on spheres), with the Fano condition, the evaluation image of such strata is always of codimension of two or higher. Hence, it is plausible that these moduli spaces with evaluation maps define currents on L. But to make this precise seems to be a non-trivial problem. A similar problem also has been observed in the case of Gromov-Witten theory if one tries to integrate forms over the pseudo-cycle (see p. 277 of [MS]). The author does not know how to prove it, so we require the following strict assumption on the sympletic manifold so that the moduli chain defines a current. Assumption 3.1. The toric Fano manifold M is assumed to be convex. Namely we require that for any genus 0 stable map f : → M, f ∗ TM is generated by global sections. Such an assumption holds in the case of complex projective spaces, and products of complex projective spaces. Except of this rectifiability problem of the compactified moduli chain of holomorphic discs, the results in this paper hold for all toric Fano manifolds. Even when the assumption is not satisfied, the results in Sect. 6 can be understood independently as computations of some invariants (see Proposition 6.5). We remark about perturbing the standard complex structure to a tame almost complex structure. McDuff and Salamon [MS] showed that for a subset Jreg (M) of second category, the moduli spaces of simple J -holomorphic curves become pseudo-cycles. In
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the case of J-holomorphic discs, it is more complicated since the structure of non-simple J-holomorphic discs can be very complex. But due to the structure theorem proven by Kwon and Oh [KO], a similar proof as in [MS] can be used to show that the moduli space of simple discs are “pseudo-chain” which may be similarly defined as pseudo-cycle. But also in this case, we do not know if these moduli chains would define currents. If these define currents, one can prove the invariance of Floer cohomology ring in a similar way as in [FOOO]. Another approach would be to consider the Kuranishi structure of the moduli space of J -holomorphic discs ([FOOO, FOno]). But as pointed out in [FOOO], it is not (yet) possible to find a Kuranishi perturbation which is compatible for all homotopy classes in π2 (M, L). Such compatibility is rather essential since we are interested in the relations between moduli spaces which produce A∞ -formula. 3.2. Transversal A∞ -algebra. Now we explain how to achieve transversality of the fiber product in the definition of A∞ -formulas. First, recall that the ordinary intersection product in the chain level is not well-defined, while the cup product is well-defined on cohomology. Hence, even in the classical level, the A∞ -algebra (C ∗ (L; nov ), mk,0 ) is not easy to define, since operations are defined in the chain level. But it is obvious how to define it to work only transversally. A similar problem occurs for mk,β . For example the fiber product mk (P , P , . . . , P ) is not transversal if P = L. Hence, the authors of [FOOO] develop a non-trivial technique to overcome such a problem. In this section, we show that if we choose the generic sequence of chains, then the fiber product is transversal, and this transversal A∞ -algebra is enough to determine homology and its ring structure. Definition 3.2. A k-tuple (P1 , . . . , Pk ) is called a transversal sequence if the chain (P1 × · · · × Pk ) is transversal to the image of the map evβ for all β ∈ π2 (M, L). For a transversal sequence (P1 , . . . , Pk ), the fiber product mk (P1 , . . . , Pk ) is well-defined. Recall that a residual subset of a space X is one which contains the intersection of countably many dense open subsets. Lemma 3.2. For a residual set of C ∗ (L; nov )×· · ·×C ∗ (L; nov ), the k th A∞ -formula (2.1) is well-defined. Namely all the fiber products given in the formula are transversal. Proof. It is enough to show that transversality of the chain (P1 × · · · × Pk ) and the image of evβ from each codimension 1 strata of the moduli space of J -holomorphic discs for all β ∈ π2 (M, L), which can be achieved by choosing generic chains Pi ’s by the standard transversality theorem.
Corollary 3.3. (C ∗ (L; nov ), {mk }) satisfies the A∞ -formula for a dense transversal sequence of chains. In fact, in our case it is easy to perturb (P1 , . . . , Pk ) to a transversal sequence due to the presence of torus action. Namely, as the torus (S 1 )n acts on L transitively. Hence, for a generic (t1 , . . . , tk ) ∈ (S 1 )n × · · · × (S 1 )n , (t1 · P1 ) × · · · × (tk · Pk ) is a transversal sequence. Also because we have the same torus action on the moduli space of holomorphic discs, we have the following identity: mk (t · P1 , . . . , t · Pk ) = t · mk (P1 , . . . , Pk ).
(3.1)
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Therefore, the transversality of the A∞ -formula also can be achieved by the torus action on each chain: If the mk2 term causes non-transversality to define mk1 in the A∞ -formula, then we can perturb all chains inside mk2 by the same t ∈ (S 1 )n to make the mk2 term transversal in mk1 by the equality (3.1). Also, it is easy to perturb a Floer-cycle in its cohomology class by the following lemma. Lemma 3.4. Let be the chain map constructed in Definition 4.4. For any cycle P of singular homology, (P ) is a Floer-cycle, i.e. m1 ((P )) = 0. Then for t ∈ T n , t ·(P ) is also a Floer cycle, and we have (P ) − t · (P ) = (−1)n m1 (H ), where the homotopy H is a singular chain with m1,0 (H ) = P − t · P . Proof. Equation (3.1) for k = 1 implies that m1,β (t · P ) = t · m1,β P . Hence the theorem follows. The last statement follows by applying Proposition 4.1 (1) for the chain H with the fact that t · (P ) = (t · P ).
Proposition 3.5. m 2 defines a product on the Floer cohomology ring H F BM (L; J0 ). Proof. To show that the product is well-defined on cohomology, it is enough to show that for P , Q ∈ C ∗ (L; nov ) with m1 (P ) = m1 (Q) = 0, we have m2 (P , t1 · Q) = m2 (P , t2 · Q) + m1 (R) for generic t1 , t2 ∈ (S 1 )n and for some R ∈ C ∗ (L; nov ). First, for any homotopy class β ∈ π2 (M, L), note that the the fiber product in m1,β (P ) of A∞ -algebra is transversal for any chain P since the evaluation map from the moduli space is always submersive due to the torus action. And m2,β (P , Q) is transversal if m1,β (P ) is transversal to Q. Then, for a generic t ∈ (S 1 )n , m1,β (P ) is transversal to t · Q for any β. Also, for generic t1 , t2 ∈ (S 1 )n , m1,β (P ) is transversal to H with m1 (H ) = t1 · Q − t2 · Q for any β. If not, we can perturb t1 · Q, t2 · Q, H by another t ∈ (S 1 )n to make them transversal. Therefore, m2 (P , t1 · Q) − m2 (P , t2 · Q) = m2 (P , m1 (H )) = ±m1 (m2 (P , H ). This finishes the proof.
4. Bott-Morse Floer Cycles In [C] and [CO], Oh and the present author have shown that for any such torus fiber L ⊂ M, the Floer homology group H F (L, L) when nonvanishing, is isomorphic to the singular cohomology of the Lagrangian submanifold H ∗ (L : nov ). Now, we fix a Lagrangian torus fiber L whose Floer cohomology is non-vanishing. The fact that H F BM (L; J0 ) and H ∗ (L; nov ) is isomorphic as a module is a little bit deceiving because a cycle in the singular homology is not a cycle in Floer homology. We need to modify a cycle, say P , by adding correction terms, say Q to make it satisfy m1 (P +Q) = 0. In the computations of [C] or [CO], it was automatically taken care of by the spectral sequence. We will find exact correction terms for any cycle in Proposition 4.1. Actually
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we will construct a filtered chain map from the singular chain complex to the Bott-Morse Floer complex. We start with the following definition and an important example to understand the construction that follows. Definition 4.1. An element P = ki=1 ai [Pi , fi ] T ei q µi ∈ C ∗ (L; nov ) is called a Floer-cycle if m1 (P ) = 0. Example 4.2. Consider a Clifford torus T 2 in CP 2 . A point < pt > is a cycle in the singular homology of T 2 . Let l0 , l1 , l2 be the cycles in T 2 which are boundaries of holomorphic discs [z; 1; 1], [1; z; 1], and [1; 1; z]. These three discs have the same symplectic area which we denote by ω(D). Recall from [C] that we have m1 < pt >= (−1)n (l0 + l1 + l2 ) ⊗ T ω(D) q = 0. Therefore < pt > is not a Floer-cycle. But, l0 + l1 + l2 is homologous to zero. We may choose a 2-chain Q ⊂ L with ∂Q = −(l0 + l1 + l2 ). Hence < pt > +Q ⊗ T ω(D) q turns out to be a correct Floer-cycle: m1 (< pt > +Q ⊗ T ω(D) q) = m1,2 (< pt >) + m1,0 (Q) ⊗ T ω(D) q = (−1)n ((l0 + l1 + l2 ) + ∂Q) ⊗ T ω(D) q = 0.
(4.1)
Similarly, we can explicitly construct correction terms as follows for the general toric Fano case. We first recall the usual product structure on the torus T n = (S 1 )n , i.e. for (a1 , . . . , an ) ∈ T n , (b1 , . . . , bn ) ∈ T n , we have (a1 , . . . , an ) × (b1 , . . . , bn ) = (a1 b1 , . . . , an bn ). Also for subsets P ⊂ T n , Q ⊂ T n , we denote by P × Q P × Q := {(p × q) ∈ T n |p ∈ P , q ∈ Q}. We may assign the set P × Q a product orientation. Recall from [CO] that we have N holomorphic discs of Maslov index 2 (up to Aut (D 2 )) with boundary on the Lagrangian torus fiber L ⊂ M, which we denote by D1 , . . . , DN . We denote the homotopy classes of such discs as β1 , . . . , βN . Then we have m1,βi (P ) = (−1)n (∂Di ) × P .
(4.2)
Now, we recall the partition {1, 2, . . . , N} =
l
Ii
i=1
with respect to the symplectic energy of discs, i.e. discs Dj for j ∈ Ii have the same symplectic area, which we denote as ei . Nonvanishing of Floer cohomology was shown to be equivalent to the following equality for each i = 1, . . . , l: ∂Dj = 0 in H ∗ (T n ). j ∈Ii
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Definition 4.3. For each i, we denote by Qi a 2-chain with the following property. ∂Qi = −
(4.3)
∂Dj .
j ∈Ii
We may choose such a 2-chain since RHS is homologus to zero. Now, consider the chain complex C ∗ (L; nov ) defined in (2.3) with two different coboundary operators m1,0 and m1 . To distinguish the two chain complex, we label them as (C1∗ (L, nov ), m1,0 ), whose cohomology is isomorphic to singular cohomology, and (C2∗ (L, nov ), m1 ), whose cohomology is a Bott-Morse Floer cohomology. Now we define a chain map between these two complexes when Floer cohomology is non-vanishing. Definition 4.4. Let P ⊂ L be any singular chain. Define (P ) : = P +
l
(Qi × P ) ⊗ T ei +
i=1
⊗T ei +ej q 2 + · · · + ⊗T
(Qi × Qj × P )
i<j
(Qi1 × · · · × Qik × P )
i1 <···
k
j =1 eij
q k + · · · + (Q1 × Q2 × · · · × Ql × P ) ⊗ T
l
i=1 ei
ql.
By extending linearly over C ∗ (L; nov ), we obtain a map : C1∗ (T n ; nov ) → C2∗ (T n ; nov ). Remark 4.5. For simplicity, we define for singular chains rather than geometric chains. It can be easily modified to the latter case. We also recall that C ∗ (L; nov ) has a filtration with respect to energy:
F λ0 C ∗ = {
ai [Pi , fi ]T λi q mi |λi ≥ λ0 for all i}.
i
Proposition 4.1. Let L be a Lagrangian torus fiber in toric Fano manifolds, whose Floer cohomology is non-vanishing. Then, the map defines a filtered chain map which induces an isomorphism on cohomology. : H ∗ (L; nov ) → H F BM (L; J0 ). More precisely, (1) (m1,0 P ) = m1 (P ), (2) (F λ (C1 )) ⊆ F λ (C2 ). Remark 4.6. Note that is only defined when Floer homology is non-vanishing since otherwise we can not find chains Qi in 4.3.
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Proof. The second property is clear from the definition, hence we only prove the first statement, which we prove by direct calculation. Recall that m1,k ≡ 0 for k ≥ 4 in toric Fano case (see Proposition 7.2 of [CO]). Hence, m1 ((P )) = m1,0 (P ) + m1,2 (P ).
(4.4)
The first component can be written as m1,0 (P ) = (−1)n ∂(P ) = (−1)n (∂P +
l
∂(Qi × P ) ⊗ T ei + · · · )
i=1
= (−1)n ((∂P ) +
l
∂(Qi ) × P ⊗ T ei
i=1
(∂(Qi × Qj ) × P ) ⊗ T ei +ej q 2 + · · · ). + i<j
We used the following formula in the last equality, where there is no sign contribution since Qi ’s are 2-chains: ∂(Qi1 × · · · × Qik × P ) = (Qi1 × · · · × Qik ) × ∂P +
k
(Qi1 × · · · (∂Qij ) × Qik ) × P .
j =1
For the second component in (4.4), m1,2
(Qi1 × · · · × Qik−1 × P ) ⊗ T
i1 <···
=
N
m1,βj
j =1
=
=
q
k−1 l=1
eil k
q
(Qi1 × · · · × Qik−1 × P ) ⊗ T ej T
k−1 l=1
i1 <···
= −(−1)n
eil k−1
(Qi1 × · · · × Qik−1 × P ) ⊗ T ej T
−(−1)n ∂Qi ×
i=1
l=1
i1 <···
l
k−1
k
(Qi1 × · · · (∂Qij ) × Qik × P ) ⊗ T
i1 <···
(−(−1)n )∂(Qi1 × · · · × Qik ) × P ⊗ T
k
l=1 eil
k
l=1 eil
qk
qk.
i1 <···
In the third equality, we used the identity (4.2), (4.3). Hence, we have m1 ((P )) = m1,0 (P ) + m1,2 (P ) = (−1)n (∂P ) = (m1,0 P ).
eil k
q
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The arguments in this section (hence of the whole paper) can be extended to the case with different spin structures. Extension to the case with flat bundles over a Lagrangian submanifold is possible in the case that non-vanishing Floer cohomology occurs when for each i = 1, . . . , l the holonomies along discs Dj are equal for all j ∈ Ii so that we can define Qj . This includes all the examples we showed in the last section. 5. A Direct Computation of Ring Structure Now, we provide two different computations of Floer cohomology rings of torus fibers in toric Fano manifolds. In this section, we give a direct computation using the classification of holomorphic discs by Oh and the author in [CO]. For simplicity, we carry out calculations for degree 1 generators, which is enough to see the whole algebraic structure of the ring due to associativity. First we choose the generators Ci of H 1 (L) for i = 1, . . . , n. Definition 5.1. Let li be a circle 1 × · · · S 1 · · · × 1, where S 1 is the i th circle of (S 1 )n ⊂ (C∗ )n . Then torus action of (S 1 )n on L gives corresponding cycles in L, which we also denote as li by abuse of notation. For i = 1, . . . , n, denote by Ci ∈ H 1 (L) the Poincar´e dual of the cycle (−1)i−1 (l1 × · · · × lˆi × · · · ln ). Similarly, we denote by Ci,j ∈ H 2 (L) the Poincar´e dual of the cycle (l1 × · · · × lˆi × · · · × lˆj × · · · ln ) for i = j , and we also define Ci1 ,...,ik ∈ H k (L) similarly for the index set {i1 , i2 , . . . , ik }. Now we show that Ci ’s generate the Floer cohomology ring H F BM (L; J0 ). Proposition 5.1. Let L be a Lagrangian torus fiber whose Floer cohomology group H F BM (L; J0 ) is nonvanishing, thus isomorphic to H ∗ (L; nov ). Then, for each i, Ci is a Floer-cycle without any correction terms, and Floer cohomology H F BM (L; J0 ) is generated by Ci for i = 1, . . . , n as a ring. Proof. From the construction in Definition 4.4, any correction term added to Ci , like P D(Ci ) × Qj , is supposed to have chain dimension n + 1 or higher. Hence, as a current in L, they are zero. Hence, Ci itself is a Floer-cycle. To see that {Ci } generate the Floer cohomology ring, note that m2 (Ci1 , m2 (Ci2 , . . . , m2 (Cik−1 , Cik ) · · · ) is a Floer cycle whose index zero part is m2,0 (Ci1 , m2,0 (Ci2 , . . . , m2,0 (Cik−1 , Cik ) · · · ). Since m2,0 is nothing but the cup product, hence the latter equals Ci1 ,...,ik ∈ H k (L) up to sign. Note that all the other terms (terms containing m2,β with non-zero β) are higher order terms with respect to the filtration by T . Hence, these elements generate the ring H F BM (L; J0 ).
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Remark 5.2. For the sign convention for the cup product, see [FOOO] Convention 25.14. Now, we compute the quantum contribution. We first state the following lemma which is a special case of Proposition 2.1. Lemma 5.2. Let β ∈ π2 (M, L) be a homotopy class. Then the degree (as a cochain) of m2,β (Ci , Cj ) is given by deg(Ci ) + deg(Cj ) − µ(β) = 2 − µ(β). Hence, we have a non-trivial m2,β product between the generators Ci for β with µ(β) = 0 or 2. The product when µ(β) = 0 is the classical cup product, hence we consider the contributions from homotopy classes with Maslov index two. Let us recall the definition of m2,β , m2,β (Ci , Cj ) = (−1)n+1 ((Mmain (βk ) ev1 ,ev2 × (Ci × Cj ), ev0 ), 3
(5.1)
where i is an embedding of cycles into L. Recall that the main component is one of the components of the moduli space of discs with marked points ev0 , ev1 , ev2 which lie on the disc counter-clockwise direction. The fact that we use only the main component of the moduli space is important, and this makes computation a little cumbersome. To get an intuitive idea about calculations, we first study the case of CP 1 . 5.1. Example : the equator L ⊂ CP 1 . Let L be the equator of CP 1 , whose Floer cohomology H F ( L, L) is isomorphic to H ∗ (S 1 ). We pick a point p which will be an element of both the singular homology H0 (L) and H F 1 (L, J0 ). Note that the cup product P D(p) ∪ P D(p) = 0, since generically two points do not intersect in S 1 . In our case, we choose t ∈ S 1 which is not equal to 1, and consider two points p and q = t · p. Then, clearly m2,0 (p, q) = 0. Now, we consider products m2,β with non-zero β. By Lemma 5.2, we only consider β with µ(β) = 2. Recall from [C] that there are only two such holomorphic discs Du , Dl (up to Aut (D 2 )) with boundary on L, which are nothing but discs covering the upper(lower)-hemisphere Du (Dl ). Note that both discs intersect p and q. Then, the product m2,Du (p, q) is a certain part of the boundary of Du . More precisely, since we only consider the “main” component of the boundary, where ev0 , p, q is ordered counter-clockwise on the boundary of the disc Du , we obtain a part of S 1 as in Fig. 2. And similarly, the product m2,Dl (p, q) only takes the “main” component of the boundary, where ev0 , p, q is ordered counter-clockwise on the boundary of the disc Dl . Therefore, after adding these two pieces, we obtain the whole equator: m2 (p, q) = (m2,Du (p, q) + m2,Dl (p, q))T ω(D) q = [S 1 ]T ω(D) q. Hence, H F (L, J0 ) is a Clifford algebra with a generator [p] and a unit 1 = [S 1 ] such that m2 (p, q)] = [m2 (p, q)] m 2 ([p], [p]) = [ = [S 1 ]T ω(D) q = 1 · T ω(D) q ∈ H F (L, J0 ).
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Fig. 2. Main components of evaluation maps for Du (left) and Dl (right)
5.2. Computation of m2 (Ci , Cj ) + m2 (Cj , Ci ). The previous example illustrates that the product m2 (Ci , Cj ) is a sum of chains in L. Actually this is not a cycle of singular homology in general as seen in Definition 4.4. But we can make the calculation much easier by computing the sum m2 (Ci , Cj ) + m2 (Cj , Ci ) instead of each individual piece. (The next section generalizes this observation.) The reason that we compute this sum rather than each part, is that by adding each “main” component we will obtain the whole boundaries of the discs which intersect both Ci and Cj . And computing this sum will be enough to show that the algebra we obtain is a Clifford Algebra. Again the only nontrivial m2,β (Ci , Cj ) will come from the homotopy classes β1 , . . . , βN of Maslov index 2 by Lemma 5.2. We recall the relevant fiber product orientation from [C]. This in fact provides the same orientation as in [FOOO], which is described by the orientation of fiber products of Kuranishi structures. (The smooth simplex may be considered as a weakly submersive strongly continuous map from a space with Kuranishi structure with corners where the obstruction bundle is taken to be the normal bundle of the embedding). Definition 5.3 [C]. Let X, P , Y be an oriented smooth manifold. Let f : X → Y and i : P → Y be a smooth map. We define the orientation of the fibre product X ×Y P for the case that the map i : P → Y is an embedding. Let f : X → L be a submersion and i : P → L be an embedding. Here we will regard P as a submanifold of L. By x, l, p we denote the dimension of X, L, P . Take a point q ∈ f (X) ∩ P . We can choose an oriented basis < u1 , . . . , ul >∈ Tq L and < w1 , . . . , wp >∈ Tq P which agrees with the given orientations of L and P . Since f is a submersion, we can choose < v1 , . . . , vl >∈ Tp X for some p ∈ f −1 (q) such that (df )p (vk ) = uk for k = 1, . . . , l. Then, we can choose a basis < η1 , . . . , ηx−l >∈ Ker(dfp ) such that < η1 , . . . , ηx−l , v1 , . . . , vl , > is the given orientation of Tp X. Then we define an orientation on the fibre product X f ×i P so that < η1 , . . . , ηx−l , w1 , . . . , wp > becomes an oriented basis. From now on, [ ] means the oriented frame on its tangent bundle. We remark that we mainly follow the amazing work of the orientation convention in [FOOO]. We may rewrite the following [FOOO]. m2,βk (Ci , Cj ) + m2,βk (Cj , Ci ) = (−1)n+1 ((M3 (βk ) ev1 ,ev2 × (Ci × Cj ), ev0 ) = (−1)((M3 (βk ) ev1 ×i Ci ) ev2 ×i Cj ), ev0 ). Recall that k )] × [∂D0 ] × [∂D1 ] × [∂D2 ])/P SL(2 : R) [M3 (βk )] = ([M(β k )] × [∂D1 ])/P SL(2 : R) = (−1)([∂D0 ] × [∂D2 ] × [M(β = (−1)([∂D0 ] × [∂D2 ] × [T n ]).
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Here the last equality follows from [C] Proposition 3.18. By the above definition of fiber product orientation, we have [M3 (βk ) ev1 × Ci ] = (−1)[∂D0 ] × [∂D2 ] × [Ci ]. Therefore, m2,βk (Ci , Cj ) + m2,βk (Cj , Ci ) = ([∂D0 ] × [∂D2 ] × [Ci ]) ev2 ×i [Cj ]). As the marked point travels around the 3rd marked point ∂D2 , its trajectory in L is vk1 l1 + · · · + vkn ln . Here, vk for k = 1, . . . , N are normal vectors to the codimension 1 faces of the moment polytope for M, [∂D2 ] × [Ci ] = [vk1 l1 + · · · + vkn ln ] × (−1)i−1 [l1 × · · · × lˆi × · · · × ln ] = (−1)i−1 [vki li × l1 × · · · × lˆi × · · · × ln ] = vki [l1 × · · · × ln ] = vki [T n ]. Therefore, m2,βk (Ci , Cj ) + m2,βk (Cj , Ci ) = ([∂D0 ] × [∂D2 ] × [Ci ]) ev2 ×i [Cj ]) = ([∂D0 ] × [vki T n ]) ev2 ×i [Cj ] = vki ([∂D0 ] × [Cj ]) = vki vkj [T n ] = vki vkj · 1. Also note that signs of the following cup product works as m2,0 (Ci , Cj ) = −m2,0 (Cj , Ci ). Therefore, Proposition 5.3. m2 (Ci , Cj ) + m2 (Cj , Ci ) =
N
(m2,βk (Ci , Cj ) + m2,βk (Cj , Ci ))T ek q
k=1
=
N
vki vkj T ek q.
k=1
Now we consider the case when i = j . The above formula also works for the case i = j after we perturb Ci by a torus action to t · Ci for some t ∈ T n . Also we have the following easy lemma. Lemma 5.4. [m2 (Ci , t · Ci )] = [m2 (t · Ci , Ci )] in H F ∗ (L, J0 ).
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C.-H. Cho
Corollary 5.5. N 1
m2 (Ci , t · Ci ) =
k=1
2
2 ⊗ T ek q. vki
Now, we recall the definition of the Clifford algebra. Definition 5.4. Let V be a Q-vector space with a non-degenerate symmetric bilinear form Q on V . The Clifford Algebra Cl(V , Q) is defined as Cl(V , Q) = T (V )/I (V , Q), where T (V ) is the tensor algebra T (V ) =
V k,
k=0
and I (V , Q) is the ideal in T (V ) generated by elements 1 v ⊗ v − Q(v, v)1 for v ∈ V . 2 Alternatively, one may define Cl(V,Q) with the relation v · w + w · v = Q(v, w). In our case, we consider a universal Novikov ring nov instead of Q as a coefficient. Now Proposition 5.3 and Corollary 5.5 imply our main theorem. Theorem 5.6. Let L ⊂ M be a Lagrangian torus fiber in the Fano toric manifold whose Floer cohomology is non-vanishing. Then, the Floer cohomology ring (H F BM (L; J0 ), m 2 ) has a Clifford Algebra structure with generators given by Ci for i = 1, . . . , n, and its relations as 2 (Cj , Ci ) = Q(Ci , Cj ), m 2 (Ci , Cj ) + m where the symmetric bilinear form Q is given by Q(Ci , Cj ) =
N
vki vkj T ek q.
k=1
Furthermore, this Q agrees with the Hessian of the superpotential W ( ) of the mirror Landau-Ginzburg model of toric Fano manifold (upon the substitution “T 2π = e−1 ”). Proof. We only need to check the last statement. Recall that the superpotential is given as (see for example [HV, CO]) W ( ) =
N k=1
e−yk −< ,vk > .
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Hence, it is easy to see that ∂W ( ) =− vki e−yk −< ,vk > , ∂ i N
k=1
and ∂ 2 W ( ) = vki vkj e−yk −< ,vk > . ∂ i ∂ j N
k=1
Here is a the coordinate on the mirror Landau-Ginzburg model, and it is related to the toric manifold M as follows. The real part of the variable is given by (a1 , . . . , an ) ∈ P , which is the image point of the Lagrangian torus fiber L in the moment polytope P , whereas the imaginary part is given by the holonomy of the flat line bundle along L. When the Floer cohomology of L is non-vanishing, the corresponding becomes the critical point of W as shown in [CO], and 2π times its exponent (yk + < , vk >) becomes the area of holomorphic discs which we denoted as ek in this paper. If we ignore harmless grading q, and with the equivalence “T 2π = e−1 ”, e−yk −< ,vk > = T ek . Hence, this proves the claim.
More correspondences will be given in the next section. 6. Analogue of Divisor Equation for Discs In this section, we introduce an analogue of the divisor equation and this will explain how Clifford algebra structure naturally arises for Floer cohomology rings of Lagrangian submanifolds, as this section provides the alternative proof of results in the previous section. To state the result, it is better to write down the formula in terms of the L∞ -algebra (strong homotopy Lie algebra) maps. Recall that every A∞ -algebra has an underlying L∞ -algebra structure by the following relation (this is similar to the fact that the commutator of an associative algebra A defines a Lie algebra on A). Theorem 6.1 ([LM, LS](or see [Fu2])). An A∞ -structure {mk : ⊗k V → V } on the graded vector space V induces an L∞ -structure {lk : ⊗k V → V }, where for all nonnegative integer k, β ∈ π2 (M, L), (−1)(σ ) mk,β (vσ (1) ⊗ · · · ⊗ vσ (k) ), (6.1) lk,β (v1 ⊗ · · · ⊗ vk ) = σ ∈Sn
with (σ ) =
(deg(vi ) + 1)(deg(vj ) + 1).
i,j with i<j,σ (i)>σ (j )
Namely, the lk map is a skew-symmetrization of the mk map. For example, l2,β (x, y) = m2,β (x, y) + (−1)(x+1)(y+1) m2,β (y, x). The following is the Divisor equation of Gromov-Witten invariants. For a general equation involving gravitational descendents, see [H1].
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C.-H. Cho
Proposition 6.2 ([KM]). Let M be a convex algebraic manifold, For α ∈ H2 (M), let M Ig,m,α : H ∗ (V )⊗n → H ∗ (M g,n ) be the Gromov-Witten invariants. Then, for γ1 ∈ H 2 (M), and πn : M g,n → M g,n−1 , we have M M (γ1 ⊗ · · · ⊗ γn ) = (α · γ1 )Ig,n−1,α (γ2 ⊗ · · · ⊗ γn ). πn∗ (Ig,n,α
Now, we state an analogue of the divisor equation for discs. Proposition 6.3. If Pi is a cycle of cohomology degree 1 in L, then for k ≥ 1, i , . . . , Pk ), lk,β (P1 , . . . , Pk ) = (Pi · ∂β) lk−1,β (P1 , . . . , P
(6.2)
i means that Pi term is omitted. where ∂ : π2 (M, L) → π1 (L), and P Remark 6.1. Here is the sign convention for the intersection of two chains P , Q of complementary degree in L. At each transversal intersection p ∈ P ∩ Q, for a basis [Tp P ] of tangent space Tp P , and similarly for [Tp Q] and [Tp L], if [Tp P ][Tp Q] has the same orientation as [Tp L] then it is counted as (+1), otherwise it is counted as (−1). Before we prove the proposition, we show how to prove the results in the previous section using the analogue of the divisor equation for discs. Recall that by βk ∈ π2 (M, L) for k = 1, . . . , N, we denote the homotopy class of a holomorphic disc of Maslov index two corresponding to N codimension one facets of the moment polytope([CO]). By definition, we have l0,βk = m0,βk = T ek q, l1,βk (P ) = m1,βk (P ). For degree 1 generators Ci , Cj of H ∗ (L) which are defined in Definition 5.1, we apply the divisor equation for discs repeatedly l2,βk (Ci , Cj ) = (Ci · ∂βk )l1,βk (Cj ) = (Ci · ∂βk )(Cj · ∂βk )l0,βk = (vki )(vkj ) ⊗ T ek q. The last equality follows from the definitions that Ci = (−1)i−1 (l1 × · · · × lˆi × · · · ln ), ∂βk = vk1 l1 + · · · + vkn ln . Hence, it is easy to see that Ci · ∂βk = (−1)n vki . Hence, we obtain Lemma 5.3, as we have l2 (Ci , Cj ) = m2 (Ci , Cj ) + m2 (Cj , Ci ). In general, we have
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Corollary 6.4. For any Lagrangian torus fiber L in the toric Fano manifold M( whose Floer cohomology may be vanishing), we have lm (Ci1 , . . . , Cim ) =
N
lm,βk (Ci1 , . . . , Cim )
k=1
= (−1)nm
N
vki1 · · · vkim ⊗ T ek q
k=1
= (−1)(n−1)m
∂ m W ( ) , ∂ i1 · · · ∂ im
where W ( ) is the superpotential of the Landau-Ginzburg mirror model of M. This corollary extends the correspondence observed in [CO], m0 = l0 = W ( ). Note that such correspondence, considered at every Lagrangian torus fiber with flat line bundles, may be used to recover the superpotential W ( ) of the Landau-Ginzburg mirror. But the above corollary indicates that in fact one Lagrangian torus fiber with a fixed flat line bundle (whose Floer cohomology may be vanishing) in M is enough to recover the superpotential in this case: It is because the superpotential is a holomorphic function on (C∗ )n and all its partial derivatives at the corresponding point on the mirror is given from the products of the L∞ -algebra by the above correspondence. Also note that the above product does not depend on the choice of cycles C∗ since it is determined by the intersection numbers which only depend on the homology class of C∗ . These are also invariants with respect to the change of an almost complex structure. By J0 we denote the standard complex structure of the toric Fano manifold M, and J0 denote the corresponding lm products by lm . Proposition 6.5. Let L be any Lagrangian torus fiber of the toric Fano manifold M. Let J1 ∈ Jreg (M) be a tame almost complex structure such that all simple J -holomorphic discs are Fredholm regular. Then, for k = 1, . . . , N, we have J0 J1 lm,β (Ci1 , . . . , Cim ) = lm,β (Ci1 , . . . , Cim ) k k
in H ∗ (L; nov ). Proof. As in [MS], one can prove that the subset Jreg (M) is of the second category and path connected. Since any J -holomorphic disc with Maslov index two is simple and its homotopy class βk is minimal, the moduli space M(βk ; Jt ) of Jt holomorphic discs is in fact a manifold without boundary. Then, by choosing a path Jt ∈ Jreg (M), we set Mm+1 (βk ; J ) = ∪t∈[0,1] {t} × Mm+1 (βk ; Jt ) . Then we have m ∂ Mm+1 (βk ; J )ev × ( Cim ) = M(βk ; J1 )ev j =1
×(
m j =1
which proves the proposition.
Cim ) − M(βk ; J0 )ev × (
m j =1
Cim )
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Now we begin the proof of Proposition 6.3 Proof. A rough idea is that if Pi is a cycle of codimension 1, then it always intersects with the boundary of a J -holomorphic disc of homotopy class β with (Pi · ∂β) number of times (counted with sign). Hence, if Pi is dropped from the argument of mk , the resulting image should be the same up to a multiple of the intersection number. While this is the same idea as the “divisor equation” in Gromov-Witten theory, there are a few differences. First, the mk map records only part of the boundaries of J-holomorphic discs as it is defined by using only the main component Mmain . But note that intersection of Pi k and the disc may occur at an arbitrary point of the domain ∂D 2 . Hence we consider the L∞ -algebra map, lk , which will be shown to record the whole boundaries of discs. Then, the next step involves delicate sign analysis in the case that the parameter Pi is dropped i , . . . , Pk ) for a codimension 1 cycle from the mk (P1 , . . . , Pk ) to obtain mk−1 (P1 , . . . , P Pi . 2 Suppose there exists an element ((D 2 , z), h) ∈ Mmain k+1 (β), where h : D → M is a J -holomorphic map. We also assume that for fixed chains P1 , . . . , Pk in L, we have h(zi ) ∈ Pi for each i = 1, . . . , k. Boundary marked points z1 , . . . , zk (z0 is omitted here) separate ∂D 2 into k connected pieces. And only the component between the k th and 1th marked point contributes to the chain mk (P1 , . . . , Pk ), as it is obtained as an evaluation of 0th marked point which lies between those two marked point in Mmain k+1 . Now, it is easy to see that up to sign, other connected components will contribute to the chains mk (P2 , . . . , Pk , P1 ), mk (P3 , . . . , P1 , P2 ), · · · , mk (Pk , P1 , · · · , Pk−1 ), and ((D 2 , z), h) will not contribute to other terms of lk (P1 , . . . , Pk ) generically due to the ordering of marked points. Now, we show that signs in (6.1) are needed to have a coherent sign in the images of the above chains. We recall the following lemma from [FOOO]. Lemma 6.6 (FOOO, Lemma 25.3). Let σ be the transposition element (i, i + 1) in the k th symmetric group Sk . Then the action of σ on M1 (β, P1 , . . . , Pi , Pi+1 , . . . , Pk ) by changing the order of marked points is described by the following: σ (M1 (β, P1 , . . . , Pi , Pi+1 , . . . , Pk )) = (−1)(deg Pi +1)(deg Pi+1 +1) Mσ1 (β, P1 , . . . , Pi+1 , Pi , . . . , Pk ). Remark 6.2. In the first term, M1 (β, P1 , . . . , Pi , Pi+1 , . . . , Pk )) is defined by using the moduli space with boundary marked points lying cyclically, whereas in the second term, Mσ1 (β, P1 , . . . , Pi+1 , Pi , . . . , Pk ) is defined by using the moduli space Mσk with boundary marked points lying in the order z0 , . . . , zi−1 , zi+1 , zi , zi+2 , . . . , zk . Namely, in the latter case, only the labeling of two marked point is changed from the first case. Let σ ∈ Sn be a permutation denoted by (1, 2, . . . , k) (i.e. 1 → 2, 2 → 3, · · · , k → 1). Then, by applying the above lemma repeatedly, we have σ (M1 (β, P1 , . . . , Pk )) = (−1)(σ ) Mσ1 (β, P2 , . . . , Pk , P1 ), where (σ ) is the same sign as appeared in (6.1). Now, it is not hard to check that the latter has the same sign as (−1)(σ ) M(β; P2 , . . . , Pk , P1 ). Here, Mσ1 (β, P2 , . . . , Pk , P1 ) and M(β; P2 , . . . , Pk , P1 ) have different images coming from the same set of J -holomorphic discs. This is because marked points in the former case lie on the circle in the order
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0, k, 1, 2, . . . , k − 1 and in the latter case marked points lie on the circle in the order k, 0, 1, 2, . . . , k − 1. Hence, as we evaluate at 0th marked point, their images come from the neighboring connected components of ∂D 2 separated by marked points. Hence, this proves that with the sign given as in (6.1), the image of the boundary of discs can be glued in the lk map. Now, we explain the second step which computes the change of sign as the argument i , . . . , Pk ) for a codiPi is dropped from the mk (P1 , . . . , Pk ) to obtain mk−1 (P1 , . . . , P mension 1 cycle Pi . In the computation, we will calculate the i th fiber product with Pi to remove the term from the fiber product. (β), where h : D 2 → M is a J -holomorphic map. We Let ((D 2 , z), h) ∈ Mmain k i , . . . , Pk in L, we have h(zi ) ∈ Pi for each also assume that for fixed chains P1 , . . . , P i = 1, . . . , i, . . . , k. And let Pi be a cycle of codimension one in L. If [Pi ] · ∂β is not zero, then, a generic cycle Pi should intersect with h(∂D 2 ) transversally. Hence, we obtain a corresponding element ((D 2 , z ), h) ∈ Mmain k+1 (β) with h(zi ) ∈ Pi for each i = 1, . . . , k. We recall that the moduli space Mmain k+1 (β) is oriented as
[M(β)] × [∂D02 ] × · · · × [∂Dk2 ] /P SL(2; C),
where [∂Di2 ] denotes the tangent vector corresponding to the counterclockwise rotation of the i th marked point. If we take [∂Di2 ] to the last, we have 2 2 2 × [∂D02 ] × · · · [∂D = (−1)s1 [M(β)] i ] × [∂Dk ] /P SL(2; C) × [∂Di ], where s1 = k − i + 1. Now we write M(β; P1 , . . . , Pk ) = (−1)s2 Mmain k+1 (β)ev1 ,...,evk × (P1 × · · · × Pk ) s3 = (−1) · · · Mmain k+1 (β)ev1 × P1 · · ·evk × Pk , l l where s2 = (n + 1) k−1 deg(Pj ), s3 = k−1 j =1 j =1 deg(Pj ). l=1 l=1 main Then, if we look at the term Mk+1 (β)ev1 × P1 , it can be oriented as (−1)s1
2 2 2 [M(β)] × [∂D02 ] × · · · [∂D i ] × · · · × [∂Dk ] /P SL(2; C) × [∂Di ] ev
1
×[P1 ]
2 2 2 o (β)] × [∂D02 ] × · · · [∂D = (−1)s4 [M i ] × · · · × [∂Dk ] /P SL(2; C) × [∂Di ] ×[L] ev × [P1 ] 1 s4 2 2 o (β)] × [∂D02 ] × · · · [∂D = (−1) ( [M i ] × · · · × [∂Dk ] /P SL(2; C) ×[∂Di2 ] × [P1 ] 2 2 o (β)] × [∂D02 ] × · · · [∂D = (−1)s5 ( [M i ] × · · · × [∂Dk ] /P SL(2; C) × [P1 ] ×[∂Di2 ] 2 2 × [∂D02 ] × · · · [∂D = (−1)s6 [M(β)] i ] × · · · × [∂Dk ] /P SL(2; C) ev ×[P1 ] 1
×[∂Di2 ],
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o (β)][L] = [M(β)] where [M and s4 = n(k + 2) + s1 , s5 = s4 + p1 = s4 + dim(P1 ) and s6 = s5 + n(k + 1). Hence s6 = n + p1 + k − i + 1 = deg(p1 ) + k − i + 1. Now we repeat this process up to Pi−1 and · · · Mmain k+1 (β)ev1 × P1 · · · × Pi−1 is oriented as 2 2 (−1)s7 · · · ( [M(β)] × [∂D02 ] × · · · [∂D i ] × · · · × [∂Dk ] /[P SL(2; C)] ev1 ×[P1 ] × · · · × [Pi−1 ] × [∂Di2 ] with s7 = deg(P2 ) + · · · + deg(Pi−1 ) + k − i + 1. Then, · · · Mmain k+1 (β)ev1 × P1 · · · × Pi−1 ev × Pi i
is oriented as 2 2 (−1)s8 · · · ( [M(β)] × [∂D02 ] × · · · [∂D i ] × · · · × [∂Dk ] /[P SL(2; C)] ev1 o ×[P1 ] × · · · × [Pi−1 ] × [∂Di2 ][Pi ] 2 2 × [∂D02 ] × · · · [∂D = (−1)s9 · · · ( [M(β)] i ] × · · · × [∂Dk ] /[P SL(2; C)] ev1 o ×[P1 ] × · · · × [Pi−1 ] × [Pi ][∂Di2 ] 2 2 × [∂D02 ] × · · · [∂D = (−1)s10 · · · ( [M(β)] i ] × · · · × [∂Dk ] /[P SL(2; C)] ev1 ×[P1 ] × · · · × [Pi−1 ] , i−1 where s8 = j =1 deg(Pj ) + n, s9 = s8 + (n − 1) · 1, s10 = s9 + . The last equality follows from the sign of the intersection [Pi ][∂β] = (−1) [L]. Hence s10 = + (n − 1) + n + i−1 j =1 deg(Pj ) + k − i + 1. Now, the last expression can be considered as an orientation of · · · Mmain (β)ev1 × P1 · · ·evi−1 × Pi−1 . k Hence, orientation of M(β; P1 , . . . , Pk ) corresponds to (−1)s3 · · · Mmain k+1 (β)ev1 × P1 · · ·evk × Pk , i × · · ·evk × Pk , (β)ev1 × P1 · · · × P ⊂ (−1)s11 · · · Mmain k i , . . . , Pk ), = (−1)s12 M(β; P1 , . . . , P where s11 = s3 + s10 , s12 is obtained in a similar way as s3 and we have s12 = + k − i + (k − i)deg(Pi ) = , since deg(Pi ) = 1. This proves that if Pi intersect with the J -holomorphic disc at several boundary points, then at each intersection, the sign change between mk (P1 , . . . , Pk ) and mk−1 (P1 , . . . , i , . . . , Pk ) of the contribution from this J -holomorphic disc, is given by (−1) , where P
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is the sign of the intersection between [Pi ] and the J -holomorphic disc at each intersection point. Now, we prove the proposition. Note that in the expression lk (P1 , . . . , Pk ), a term i , . . . , Pk , Pi ) or any mk (P1 , . . . , Pk ) carries the same sign as the term mk (P1 , . . . , P other term which is obtained by moving Pi around if deg(Pi ) = 1. Consider the J i , . . . , holomorphic disc contributing non-trivially to the expression mk−1,β (P1 , . . . , P i , . . . , Pk ). Pk ). Then the whole boundary of this disc contributes to lk−1,β (P1 , . . . , P Generically, Pi may intersect with ∂β at arbitrary points of the domain ∂D 2 . Suppose such disc intersect Pi between j and j + 1th marked point with intersection sign (−1) for j > i + 1 without loss of generality. Then, the whole boundary of this disc would contribute to the terms (while divided into several pieces) i , . . . , Pj −1 , Pi , Pj , . . . , Pk ) mk,β (P1 , . . . , P i , . . . , Pj −1 , Pi , Pj , . . . , Pk , P1 ) +mk,β (P2 , . . . , P
i , . . . , Pj −1 , Pi , Pj , . . . , Pk−1 ). + · · · + mk,β (Pk , P1 , · · · , P
i , By applying the sign analysis, the above terms correspond to terms in lk (P1 , . . . , P . . . , Pk ) with multiplicity (−1) . By adding up all the possibilities of intersections between Pi and ∂β, we obtain the proposition.
7. Examples In what follows we omit area terms T ei q for simplicity. 7.1. The Clifford torus T 2 ⊂ CP 2 . In [CO], it is shown that the Clifford torus is the only Lagrangian torus fiber whose Floer cohomology is non-vanishing, which is isomorphic to H ∗ (T 2 ; nov ). Hence, by Theorem 5.6, H F ∗ (T 2 , T 2 ) as a ring is a Clifford algebra with two generators C1 and C2 . Using its moment polytope data, one can immediately compute the matrix of the symmetric bilinear form 2 1/2 Q= . 1/2 2 (See [KL] for computations of the B-model by physical arguments and the predictions made for the Clifford torus case.) But it is also instructive to compute m2 (C1 , C1 ) and m2 (C1 , C2 ) directly. Consider T 2 as a rectangle whose edges are glued accordingly. Let us assume that its edges are cycles l1 , l2 as given in Definition 5.1. Then by definition, we have C1 = l2 , C2 = −l1 . First we consider m2 (C1 , C1 ) = m2 (l2 , l2 ). As before, we pick t ∈ T 2 so that l2 and tl2 do not intersect. Then, m2,0 (l2 , tl2 ) = 0. Recall that there exist 3 holomorphic discs (up to Aut (D 2 )) with boundary trajectory as ∂D0 = −l1 − l2 , ∂D1 = l1 , ∂D2 = l2 .
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Fig. 3. m2,β0 (l2 , tl2 ) + m2,β1 (l2 , tl2 ) = L, m2,β0 (l1 , l2 )
For m2,β (l2 , tl2 ), holomorphic discs D0 , D1 contribute nontrivially. Since we only consider the main components as in Fig. 3, we have m2,β0 (l2 , tl2 ) + m2,β1 (l2 , tl2 ) = [L]. Therefore, we have m 2 ([C1 ], [C1 ]) = [m2 (C1 , tC1 )] = [L]T ω(D) q. This agrees with Corollary 5.5. From Fig. 3, it is easy to see that the product m2 (l2 , tl2 ) is independent of t ∈ T 2 . Now, we consider the product m2 (C1 , C2 ), m2 (C1 , C2 ) = m2,0 (C1 , C2 ) +
m2,β (C1 , C2 )T Area(β) q.
β
Here m2,0 (C1 , C2 ) is a cup product which is nothing but the Poincar´e dual of the intersection C1 ∩ C2 = point. But as we discussed in Example 4.2, the point itself is not a Floer-cycle. The needed correction term Q is obtained in this case from the quantum contribution m2,β . It is easy to see that only the β0 disc contributes to the product m2,β , since other discs generically do not intersect both C1 , C2 . Now, m2,β0 (C1 , C2 ) is not a cycle but a chain as drawn in Fig. 3, since we only evaluate on the main component. This is the chain Q that we added to make < pt > a Floer cycle in Definition 4.4, m2 (C1 , C2 ) = m2,0 (C1 , C2 ) + m2,β0 (C1 , C2 )T e q = < pt > +QT e q.
7.2. CP 1 × CP 1 . Consider CP 1 × CP 1 whose moment map image is a rectangle. For the equator S 1 ∈ CP 1 , S 1 × S 1 ⊂ CP 1 × CP 1 has nontrivial Floer cohomology and its product structure is given by Q=
20 . 02
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7.3. CP n . The example CP 2 easily generalizes to CP n . The Floer cohomology of the Clifford torus T n ⊂ CP n becomes the Clifford Algebra with n generators with symmetric bilinear form as 2 1/2 · · · 1/2 1/2 2 · · · 1/2 Q= . .. . . .. ... . . . 1/2 1/2 · · · 2
Acknowledgements. The author does not claim originality of the construction of the A∞ -algebra of Lagrangian submanifold which should be given to Fukaya, Oh, Ohta and Ono for their ingenious work. We would like to thank Yong-Geun Oh, and the referee for helpful comments, and for pointing out a mistake in the original version. Part of the paper was written during author’s visit to the Institute Mathematical Sciences Research Institute, and he would like to thanks for its hospitality.
References [C]
Cho, C.-H.: Holomorphic discs, spin structures and the Floer cohomology of the Clifford torus. Int. Math. Res. Not. 35, 1803–1843 (2004) [CO] Cho, C.-H., Oh, Y.-G.: Floer cohomology and disc instantons of Lagrangian torus fibers in toric Fano manifolds. http://arxiv.org/list/math.SG/0308225, 2003 [Fu1] Fukaya, K.: Morse homotopy, A∞ -category and Floer homologies. In: Proceedings of GARC Workshop on Geometry and Topology, Kim, H. J. (ed.), Seoul: Seoul National University, 1933 [Fu2] Fukaya, K.: Deformation theory, homological algebra and mirror symmetry. In: Geometry and physics of branes. (Como, 2001), Ser. High Energy Phys. Cosmol. Gravit., Bristol: IOP, 2003, pp. 121–209 [FOh] Fukaya, K., Oh, Y.-G.: Zero loop open strings in the cotangent bundle and Morse homotopy. Asian J. Math 1, 99–180 (1997) [FOOO] Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory-anomaly and obstruction. Kyoto University preprint, 2000 [FOno] Fukaya, K., Ono, K.: Arnold conjecture and Gromov-Witten invariants. Topology 38, 933–1048 (1999) [GP] Guillemin, V., Pollack, A.: Differential Topology. Prentice-Hall, Inc., NJ: Englewood Cliffs, 1974 [H1] Hori, K.: Constraints for topological strings in D ≥ 1. Nucl. Phys. B 439(1–2), 395–420 (1995) [H2] Hori, K.: Linear models in supersymmetric D-branes. In: Proceedings of the KIAS conference-Mirror Symmetry and Symplectic Geometry (Seoul, 2000), Fukaya, K., Oh, Y.-G., Ono, K., Tian, G., (eds.), River Edge, NJ: World Sci. Publishing, 2001 [HV] Hori, K., Vafa, C.: Mirror symmetry. http://arxiv.org/list/hep-th/0002222, 2000 [KL] Kapustin, A., Li, Y.: D-branes in Landau-Ginzburg models and algebraic geometry. JHEP 0312, 005 (2005) [KL2] Kapustin, A., Li,Y.: Topological correlators in Landau-Ginzburg models with boundaries. Adv. Theor. Math. Phys. 7, 727–749 (2004) [K] Kontsevich, M.: Homological algebra of mirror symmetry. ICM-1994 Proceedings, Z¨urich: Birkh¨auser, 1995 [KM] Kontsevich, M., Manin, Y.: Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys. 164 (3), 525–562 (1994) [KO] Kwon, D., Oh, Y.-G.: Structure of the image of (pseudo)-holomorphic discs with totally real boundary condition. Commun. Anal. Geom. 8 (1), 31–82 (2000) [LM] Lada, T., Markl, M.: Strongly homotopy Lie algebras. Comm. Algebra 23(6), 2147–2161 (1995) [LS] Lada, T., Stasheff, J.: Introduction to SH Lie algebras for physicists. Int. J. Theor. Phys. 32(7), 1087–1103 (1993) [MS] McDuff, D., Salamon, D.A.: J-holomorphic Curves and Quantum Cohomology. American Mathematical Society, Colloquium Publications 52, Providence, RI: Amer. Math. Soc. 2004 [O] Orlov, D.: Triangulated categories of singularities and D-branes in Landau-Gzinburg models. http://arxiv.org/list/math.AG/0302304, 2003
640 [Oh1] [Oh2] [S1]
C.-H. Cho Oh, Y.-G.: Floer cohomology of Lagrangian intersections and pseudo-holomorphic discs I. Comm. Pure and Appl. Math. 46, 949–994 (1993); addenda, ibid, 48, 1299–1302 (1995) Oh, Y.-G.: Floer cohomology, spectral sequence, and the Maslov class of Lagrangian embeddings. IMRN, 7, 305–346 (1996) Stasheff, J.: Homotopy associativity of H -spaces. I, II. Trans. Amer. Math. Soc. 108, 275–292 (1963); ibid. 108, 293–312 (1963)
Communicated by N.A. Nekrasov
Commun. Math. Phys. 260, 641–658 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1382-x
Communications in
Mathematical Physics
Density of the SO(3) TQFT Representation of Mapping Class Groups Michael Larsen , Zhenghan Wang Department of Mathematics, Indiana University, Bloomington, IN 47405, USA. E-mail: [email protected]; [email protected] Received: 7 December 2004 / Accepted: 24 December 2004 Published online: 28 June 2005 – © Springer-Verlag 2005
Abstract: We show that for an odd prime r > 3 and an integer g > 1, in the projective representation given by the SO(3) Witten-Reshitikhin-Turaev theory at an r th root of unity, the image of the mapping class group of a surface of genus g is dense. 0. Introduction A (2+1)-dimensional topological quantum field theory (TQFT) determines, for each g ≥ 0, a projective representation (ρg , Vg ) of the mapping class group Mg of a closed oriented surface of genus g. This paper is concerned with the SO(3) TQFT at an r th root of unity, r ≥ 5 prime. Those TQFTs were first constructed mathematically in [T]. The problem we consider is this: what is the closure of ρg (Mg )? For g = 1 and the SU(2)-theory, Kontsevich observed that the image is finite. A proof of finiteness both for the SU(2)-theory and the SO(3)-theory may be found in [G]; see also [J], for an early calculation from which the finiteness can be deduced. For g ≥ 2, the image was shown to be infinite, and its closure therefore of positive dimension [F]. In this paper, we identify the representation for g = 1 and show that the image is either SL2 (Fr ) or PSL2 (Fr ) depending on whether r is congruent to 1 or −1 (mod 4). For g ≥ 2, on the other hand, we show that the image is as large as possible—that is, it is dense in the group of projective unitary transformations on the representation space of Mg . We are working with the SO(3)-theory because the statements are cleaner than for the SU(2)-theory. Using the density result here and the tensor decomposition formulas in Theorem 1.5 [BHMV], the closed images of the SU(2)-theory can be identified. The restriction to a prime r is dictated by a result of Roberts [R] of which we make essential use: the SU(2)-theory representations of Mg are irreducible for r prime.
Partially supported by NSF DMS 0100537 and DMS 0354772. Partially supported by NSF EIA 0130388, and DMS 0354772, and ARO.
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The case r = 3 is trivial since dim Vg = 1 for all g. The case r = 5 was treated in joint work with Freedman [FLW2]. Our proof depended on the observation that a Dehn twist acts with r−1 2 distinct eigenvalues. For r = 5, this means that the Mg representations satisfy the “two-eigenvalue property.” The compact Lie groups G admitting a representation with respect to which a generating conjugacy class has only two eigenvalues can be classified, and the few possibilities can be reduced to one by an examination of the branching rules for the restriction Mg−1 ⊂ Mg and dimension computations (especially the Verlinde formula). For r ≥ 7, we no longer have the two-eigenvalue property, and the number of possibilities for G grows rapidly with r. Mainly, therefore, we depend on the branching rules. A crucial point is to prove that the representations are tensor-indecomposable, i.e., not equivalent to a tensor product of representations of lower degrees; this is precisely why the SO(3) case is simpler than that of SU(2). The reason tensorindecomposability is so important is that, coupled with irreducibility, it implies that the identity component of the closure of ρg (Mg ) is a simple group, and this greatly shortens the list of possibilities. The original motivation for this work was topological quantum computation in the sense of [FKLW, FLW1, FLW2]. As in [FLW2], there are also applications to the distribution of values of 3-manifold invariants. As a simple example, we show that the set of the norms of the Witten-Reshetikhin-Turaev SO(3) invariants of all connected 2π i 3-manifolds at A = ie 4r is dense in [0, ∞) for primes r ≥ 5. 1. The SO(3)-TQFT There are several constructions of the SU(2) and the SO(3) TQFTs in the literature (e.g., [BHMV, FK, RT, T]). The SU(2) TQFT was first constructed mathematically in [RT], and the SO(3) TQFT in [T]. We will follow Turaev’s book [T], where the construction of a TQFT is reduced to the construction of a modular tensor category. 2π i
2π i(r+1)
Fixing a prime r ≥ 5 and setting A = ie 4r = e 4r , note that A is a primitive 2r th root of unity when r ≡ 1 mod 4, and a primitive r th root of unity when r ≡ −1 mod 4. In [BHMV] to construct TQFT using the skein theory, the Kauffman variable A is either a primitive 4r th or a primitive 2r th root of unity. When r ≡ 1 mod 4 by the SO(3) TQFT we mean the TQFT denoted by Vr in [BHMV] with the above choice of A. When r ≡ −1 mod 4, the same construction still gives rise to a TQFT although A is only a primitive r th root of unity, which is also denoted by Vr here, but the decomposition formula in Theorem 1.5 [BHMV] does not necessarily hold. Consequently we have to distinguish between the two cases r ≡ 1 mod 4 and r ≡ −1 mod 4. The modular tensor categories associated to the TQFTs Vr are described in [T, Chapter XII]. In particular, [T , Theorem 9.2] discussed the unitarity of the TQFTs. For the above choice of A’s, the ribbon categories of [T , Theorem 9.2] are not modular because the S-matrices as given in [T , Lemma 5.2] are singular. (The Kauffman variable A is a primitive 4r th root of unity for only even r’s.) But it can be shown that the even subcategories (see [T , Sect. 7.5]) are indeed modular and unitary [T], FNWW]. The even subcategories correspond to the restriction of the representation categories to the odd dimensional (or integral spin) “halves” in the quantum group setting [FK]. A modular tensor category consists of a large amount of data. For our purpose here we will only specify the isomorphism classes of simple objects, called labels of the associated TQFT, the S-matrix, and the T matrix. More information is contained in Lemma 2.
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−2k
We write the quantum integers [k]A = A2k−A . The label set of the Vr theory is A2 −A−2 L = {0, 2, 4, . . . , r − 3}. The quantum dimension of the label i is given by di = [i + 1], the subscript A in [k]A will be dropped from now on, and√the global dimension of the r 2 ˜ S˜ = (˜sij ) Vr modular tensor category is D = π . The S-matrix i∈L [i + 1] = 2 sin
r
can be read off from Lemma 5.2 [T] as s˜ij = [(i + 1)(j + 1)]. The T = (tij ) matrix is diagonal with diagonal entries the twists θi , which are computed in [KL , Prop. 6, p. 43] as θi = Ai(i+2) . 0 −1 11 Let s = ,t = be the generators of SL2 (Z). It is a deep fact that the 1 0 01 S, T matrices give rise to a projective matrix representation of SL2 (Z) if we make the ˜ t → T −1 , i.e., the SO(3) TQFT representation following assignments: s → S = D1 · S, ρSO(3) for SL2 (Z) is: ρSO(3) (s) =
1 ([i + 1][j + 1]) D
and ρSO(3) (t) = (A−j (j +2) δij ). The T -matrix corresponds geometrically to a Dehn twist, so the negative twists θi−1 are the eigenvalues for the image of any Dehn twist on a non-separating simple closed curve. Remark. It seems to be generally believed that the two theories Vr and V2r constructed in [BHMV] correspond to the SO(3) and the SU(2) Witten-Reshetikin-Turaev TQFTs. Actually the S-matrix of the V2r theory is not the same as that in the Witten-ReshetikinTuraev SU(2) theory [Wi, RT]: the (i, j )th entry differs by a sign (−1)i+j . But this discrepancy disappears on restriction to the even subcategories; this is the reason that the Witten-Reshetikhin-Turaev TQFTs are always unitary, but the V2r theories are not unitary in general. This subtle point is due to the Frobenius-Schur indicators for self-dual representations, and will be clarified in [FNWW]. Lemma 1. If A is a primitive 2r th root of unity and r ≥ 3 is odd, then there exists a TQFT V2 and a natural isomorphism of theories such that V2r () ∼ = V2 () ⊗ Vr (). Moreover, the SO(3)-theory representations of the mapping class groups Mg are irreducible for all primes r ≥ 5. Proof. The decomposition formula is Theorem 1.5 of [BHMV]. To prove the second part, first consider the case r ≡ 1 mod 4. Suppose the SO(3)theory representation of the mapping class groups for a closed surface is reducible for a prime r. Then by the tensor decomposition formula above the SU(2)-theory representation would be reducible, too. But this contradicts the result of [R]. Therefore, the SO(3)-theory representations are also irreducible. For r ≡ −1 mod 4, the same argument will work if a similar decomposition formula holds. Without such a formula, the irreducibility of the SO(3) representations of Mg can be deduced by following Roberts’s argument [R].
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Lemma 2. Let dr,g,h1 ,... ,hk denote the dimension of the vector space associated by the SO(3) theory at an r th root of unity to a compact oriented surface of genus g with k boundary components labeled h1 , . . . , hk ∈ 2Z. Then we have the following: 1) dr,0,h = δh0 , where δ is the Kronecker delta. 2) dr,0,h1 ,h2 = δh1 h2 . 3) dr,0,h1 ,h2 ,h3 is 1 if and only if the hi satisfy the triangle inequality (possibly with equality) and h1 + h2 + h3 ≤ 2(r − 2); otherwise it is 0. 4) (Gluing formula) Suppose is cut along a simple closed curve, and x denotes the resulting surface with the two new boundary components both labeled by x. Then Vr () ∼ = ⊕x∈L Vr (x ), where L = {0, 2, · · · , r − 3} is the label set of the Vr theory. 5) dr,1,h = r−1−h 2 . 6) dr,g,h1 ,... ,hk ,0 = dr,g,h1 ,... ,hk . r−1 r csc2 2πj g−1 2 r αj , where αj = . 7) (Verlinde formula) dr,g = j =1 4 Proof. Parts 1)–4) are the basic data of the Vr theory (see also [KL]). Parts 5) and 6) are easy consequences of 1)–4). The Verlinde formula is derived in [BHMV]. 2. Tensor Products and Decompositions Next we prove some technical results which enable us to establish that certain representations are tensor indecomposible. We say a complex representation V of a compact Lie group G is isotypic if it is of the form W n = W ⊗Cn for some irreducible representation W of G. If W is one-dimensional, we say V is scalar. Two representations of G are conjugate if one is equivalent to the composition of the other with an automorphism of G. Lemma 3. Let 0 → G1 → G2 → G3 → 0 be a short exact sequence of compact Lie groups and ρ : G2 → GL(V ) an irreducible representation. If the restriction of V to ˜ 1 and G ˜ 2 of G1 and G2 respectively G1 is isotypic, then there exist central extensions G ˜ 2 such that and representations σ and τ of G ˜ 1 is a normal subgroup of G ˜ 2, (1) the extension G ˜ ˜ (2) the quotient G2 /G1 is isomorphic to G3 , ˜ 1 is irreducible, (3) the restriction of σ to G ˜ 1 is scalar, and (4) the restriction of τ to G (5) the tensor product σ ⊗ τ is equivalent to the composition of ρ with the central ˜ 2 → G2 . quotient map G Proof. By hypothesis, V |G1 is isotypic and so can be written as W ⊗ Ck . The span of ρ(G1 ) is ρ(G1 )C = End(W ) ⊂ End(W ⊗ Ck ) = End(V ), where End(W ) maps to End(W ⊗ Ck ) by x → x ⊗ Idk . The image ρ(G2 ) lies in the normalizer End(W )End(Ck ) of ρ(G1 )C. Thus, ρ can be regarded as a map G2 → (End(W )End(Ck ))∗ = (GL(W ) × GLk (C))/C∗ .
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Let ˜ 2 = G2 ×(GL(W )×GL (C))/C∗ (GL(W ) × GLk (C)), G k ˜ 1 the pre-image of G1 in G ˜ 2 with respect to the central quobe a central extension of G2 , G ˜ 2 → G2 , and ρ˜ the pullback G ˜ 2 → (GL(W ) × GLk (C)) of ρ. Let σ and tient map π : G τ denote the compositions of ρ˜ with the projection maps GL(W ) × GLk (C) → GL(W ) and GL(W ) × GLk (C) → GLk (C) respectively. The diagram ˜1 ˜2 G1 ← G → G ↓ ↓ ↓ ρ˜ GL(W ) = (GL(W ) × C∗ )/C∗ ← GL(W ) × C∗ → GL(W ) × GLk (C) ˜ 1 are irreducible and scalar respectively, and shows that the restrictions of σ and τ to G (5) is immediate. In the other direction, we have: Lemma 4. Let G2 be a compact Lie group and G1 a closed normal subgroup. Let V and W be irreducible representations of G2 such that V |G1 is irreducible and W |G1 is scalar. Then V ⊗ W is irreducible. Proof. As V and W are irreducible, G dim EndG2 (V ) = dim V ⊗ V ∗ 2 = 1; G dim EndG2 (W ) = dim W ⊗ W ∗ 2 = 1. Let V ⊗V∗ =
Vi , W ⊗ W ∗ =
i
Wj
j
denote decompositions into irreducible G2 -representations, numbered so that V1 and W1 are trivial (and therefore the other Vi and Wj are non-trivial). Thus, EndG2 (V ⊗ W ) =
G V ⊗ V ∗ ⊗ W ⊗ W∗ 2 = (Vi ⊗ Wj )G2 =
C.
{i,j |Vi ∼ =Wj∗ }
i,j
Now, Wj |G1 is trivial for all j , so Vi ∼ = Wj∗ implies that Vi |G1 is trivial. However, G dim ViG1 , 1 = dim EndG1 (V ) = dim V ⊗ V ∗ 1 = i
so this is possible only for i = 1. Then Wj ∼ = V1∗ implies j = 1, so dim EndG2 (V ⊗ W ) = 1.
Lemma 5. For r ≥ 5, the tensor product of any two non-trivial irreducible representations of SL2 (Fr ) has an irreducible factor of degree > r−1 2 .
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Proof. This can be deduced from the character table [Sp], p. 160, whose notation we follow. Let χ1 and χ2 be non-trivial irreducible characters of SL2 (Fr ). If χ1 χ2 = eχβ+ + f χβ− + g, then
g=
1 0
if χ2 = χ¯ 1 , otherwise.
If db ∈ T1 is a square, then χβ±1 (db ) = −1. If χ2 = χ¯ 1 , comparing values at db , e+f ≤ 1, which is absurd. Thus g = 0, and thus |χ1 (db )χ2 (db )| = e + f . As χ (1) ≥ r−1 2 |χ (db )| for all non-trivial χ , we get a contradiction. Lemma 6. Consider a short exact sequence of compact Lie groups 0 → G1 → G2 → PSL2 (Fr ) → 0, where r ≥ 7. If H2 is the inverse image of a proper subgroup H ⊂ PSL2 (Fr ), and V is r−1 2 a representation of H2 , then some irreducible factor of IndG H2 V has degree > 2 . Proof. Without loss of generality, we may assume that V is irreducible. Let K = 2 ker G1 → GL(IndG H2 V ). Replacing G2 and H2 by G2 /K and H2 /K respectively if
2 necessary, we may assume the restriction of IndG H2 V to G1 is faithful. Suppose that the
2 restriction of some irreducible factor W of IndG H2 V to G1 fails to be isotypic. By Clifford’s theorem, W is a direct sum of mutually conjugate isotypic representations Wik of G1 ; the stabilizer of W1k can be regarded as a subgroup of PSL2 (Fr ), and
dim W = k dim W1 [PSL2 (Fr ) : ]. By the well-known classification of subgroups of PSL2 (Fr ), each proper subgroup has r−1 index > r−1 2 , so dim W > 2 . We may therefore assume that the center of G1 is in the center of G2 . By a theorem of Eilenberg and Mac Lane [EM], the obstruction to finding a section of G2 → PSL2 (Fr ) lies in H 2 (PSL2 (Fr ), Z(G1 )). As PSL2 (Fr ) is perfect with universal central extension SL2 (Fr ), G2 contains a subgroup isomorphic to PSL2 (Fr ) or SL2 (Fr ) which maps onto PSL2 (Fr ). As
2 Res G2 IndG H2 V = IndH2 ∩ ResG2 V ,
we can reduce to the case that G2 is PSL2 (Fr ) or SL2 (Fr ). If G2 = PSL2 (Fr ) and H˜ 2 2 is the preimage of H2 ⊂ G2 in SL2 (Fr ), then IndG H2 V , regarded as a representation of SL2 (Fr ) ˜ V , where V˜ is V regarded as a representation of H˜ 2 . SL2 (Fr ) is the same as Ind H˜ 2
Therefore, without loss of generality, we may (and do) assume G = SL2 (Fr ). If H1 ⊂ H2 ⊂ G, H2 G IndG H2 IndH1 V1 = IndH1 V1 ,
so without loss of generality we may assume H is a maximal proper subgroup. Aside from the trivial representation, SL2 (Fr ) has two irreducible representations of dimension ± ≤ r−1 2 , with characters χβ . As V is irreducible, it is a subrepresentation of the regular
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representation of H , and it follows that IndG H V is a subrepresentation of the regular representation of SL2 (Fr ). In particular, [G : H ] dim V reduces to 0 or 1 (mod r−1 2 ), the former if dim V > 1, and [G : H ] dim V ≤ 12 + (dim χβ+ )2 + (dim χβ− )2 =
|G| r 2 − 2r + 3 < . 2 2(r + 1)
It follows that |H | > 2(r + 1). By the classification of maximal subgroups of SL2 (Fr ), this means that H is a Borel subgroup B of G or that |H | ∈ {24, 60, 120}. The irreducible representations of B all have degree 1 or r − 1. If dim V = 1, the induced representation has degree r + 1, which does not satisfy the congruence condition. If dim V = r − 1, the induced representation has degree r 2 − 1, which does not satisfy the inequality condition. This leaves a short list of possible triples (r, dim V , |H |). For r > 11, all can be ruled out by the congruence condition or the inequality condition. The only triples not ruled out are (7, 1, 48) and (11, 1, 120). In each case, the degree of the induced representation is congruent to 1 (mod r−1 2 ), so V must be trivial. By [At] pp. 3, 7, the induced representation in each case has an irreducible factor of degree r − 1. Lemma 7. Consider a short exact sequence of compact Lie groups 0 → G1 → G2 → → 0, where is SL2 (Fr ) or PSL2 (Fr ), r ≥ 5. Suppose V and W are representations of G2 . If V ⊗ W has (1) all its irreducible factors of degree ≤ r−1 2 , (2) at least one irreducible factor of degree r−1 2 which is G1 -scalar, (3) exactly one irreducible factor of degree 1, then either V or W is one-dimensional. Proof. Suppose first that V and W are irreducible and V ⊗ W satisfies hypothesis (1). By Clifford’s theorem, we can write V |G1 and W |G1 as direct sums of mutually conjugate isotypic representations Vim and Wjn respectively. Let H2 ⊃ G1 denote the subgroup n 2 m of G2 stabilizing both V1m and W1n . Then IndG H2 V1 ⊗ W1 is a G2 -subrepresentation of m ∼ V ⊗ W . By Lemma 6, it follows that V |G1 = V1 and W |G1 ∼ = W1n . (More generally, if any tensor product of G2 -representations satisfies (1), all of the irreducible constituents of all of the tensor factors are G1 -isotypic.) By Lemma 3, replacing G1 and G2 by central extensions if necessary, we can write V as Vσ ⊗ Vτ and W as Wσ ⊗ Wτ with Vσ and Wσ G1 -irreducible and Vτ and Wτ G1 -scalar. As explained above, any irreducible constituent U of Vσ ⊗ Wσ is isotypic for G1 ; passing to central extensions of G1 and G2 if necessary, we write U as Uσ ⊗ Uτ , so Uσ ⊗ (Uτ ⊗ Vτ ⊗ Wτ ) is a G2 -subrepresentation of V ⊗ W . Every irreducible factor of Uτ ⊗ Vτ ⊗ Wτ is G1 -scalar. By Lemma 5, unless at least two of Uτ , Vτ , and Wτ have dimension 1, their tensor product has an irreducible factor of degree > r−1 2 which is G1 -scalar. If dim Vτ = dim Wτ = 1, then V |G1 and W |G1 are irreducible; if X is any one-dimensional representation of G1 , then X ∗ ⊗ V |G1 is irreducible, so dim(X∗ ⊗ V ⊗ W )G1 ≤ 1. Thus, if V ⊗W additionally satisfies hypothesis (2), either dim Vτ > 1 or dim Wτ > 1. Without loss of generality, we assume the latter is true. Unless dim Uτ = dim Vτ = 1, V ⊗ W contains a factor which is the tensor product of the G1 -irreducible Uσ with a G1 -scalar irreducible of dimension > r−1 2 . This is impossible by Lemma 4. The situation
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is therefore that Vτ ⊗ Wτ is G1 -scalar, G2 -irreducible and of dimension r−1 2 , and every factor of Vσ ⊗ Wσ is G1 -irreducible. Applying (2) and Lemma 4 again, we see further that every G2 -irreducible factor of Vσ ⊗ Wσ has dimension 1, so Wσ∗ ⊗ Wσ decomposes entirely into 1-dimensional pieces over G2 . If V is any other irreducible representation such that V ⊗ W satisfies (1), then Vσ is a twist of Wσ∗ by a character so Vσ ⊗ Vσ ∗ also decomposes as a sum of (dim Vσ )2 characters over G2 . Now we return to the original problem. If V ⊗ W satisfies all three hypotheses, there exists irreducible factors V0 and W0 of V and W respectively satisfying hypotheses (1) and (2). Without loss of generality, we assume dim V0τ = 1. Then every irreducible ∗ , so they all have the same dimension d. By hypothesis (3), factor of V is a twist of W0σ there exist irreducible factors Vi and Wj of V and W respectively such that Vi ⊗ Wj has a one-dimensional G2 -submodule. Thus Wj∗ must be a twist of Vi , and (V ⊗ Wj )|G1 decomposes entirely into d 2 1-dimensional pieces. By (3), d = 1. 3. Case g = 1 Theorem 1. The projective representation of M1 = Map(S 1 × S 1 ) = SL2 (Z) given by 2π i the SO(3)-theory at A = ie 4r is the same as the projective representation obtained by composing the (mod r) reduction map SL2 (Z) → SL2 (Fr ) with the odd factor of the Weil representation of SL2 (Fr ). We want explicit matrices for one of the two r−1 2 -dimensional irreducible representations of SL2 (Fr ). To find them, we briefly recall the theory of Weil representations over finite fields [Ge]. Let Hr denote the Heisenberg group of order r 3 . We regard Hr as a central extension 0 → Fr → Hr → F2r → 0. The extension class defines a symplectic form (in this case, an area form) on the quotient. Any automorphism of Hr stabilizes the center and acts on F2r , respecting this symplectic form. Regarding SL2 (Fr ) as the group of symplectic linear transformations of F2r , we claim that its action lifts to Hr . To make this explicit, let x and y be elements of Hr whose images in F2r form a unimodular basis. Let z be the generator of the center defined by z2 = yxy −1 x −1 . Finally, for let fM (x) = zac x a y c , fM (y) = zbd x b y d , fM (z) = z. We easily check that this defines an action of SL2 (Fr ) on Hr . 2π ik Let e(k) denote e r . Let (ρ, V ) denote the Stone-von Neumann representation (i.e., the unique irreducible representation of Hr with central character zk → e(k).) We fix a basis e0 , . . . , er−1 for V so that ek is the e(2k)-eigenspace of x. In this basis, we write
1 0 0 ρ(x) = . ..
0 0 e(2) 0 0 e(4) .. .. . .
00
0
··· ··· ··· .. .
0 0 0 .. .
· · · e(−2)
,
0 0 .. . , 0 0 ··· 1 1 0 0 ··· 0
0 0 . ρ(y) = .. 0
1 0 .. .
0 1 .. .
··· ··· .. .
ρ(z) = e(1) Id.
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For any α ∈ SL2 (Fr ) ⊂ Aut(Hr ), ρ ◦α is equivalent to ρ. There exists Rα , therefore, unique up to scalar multiples, such that Rα ρ(h)Rα−1 = ρ(α(h))
(*)
for all h ∈ Hr . If R¯ α denotes the class of Rα in PGL(V ), we conclude that α → R¯ α is a projective representation. When r ≥ 5, SL2 (Fr ) is perfect and centrally closed, so there is a unique lifting to an r-dimensional linear representation, which we call the Weil representation of SL2 (Fr ). Explicitly we may choose (up to scalar multiplication) RS = (e(2ij ))0≤i,j
2
It is easy to see that the span of f0 , f2 , f4 , . . . , fr−3 forms an invariant subspace V odd of both RS and RT . In terms of this basis, RS is represented by the matrix r −1−i r −1−j r −1−i r −1−j , − e −2 RSodd= e 2 2 2 2 2 i,j ∈E and RT is represented by r −1−j 2 = δij e − 2
RTodd
.
i,j ∈E
As A2 = e
r + 1 , 2
we obtain RSodd = A2(i+1)(j +2) − A−2(i+1)(j +1)
i,j ∈E
= (A2 − A−2 ) [(i + 1)(j + 1)]A i,j ∈E ,
and RTodd = e−
2π i(r−1)2 4r
δij A−j (j +2)
i,j ∈E
.
Thus the composition SL2 (Z) → SL2 (Fr ) → PGL(V odd ) is equivalent to ρSO(3) .
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4. Case g = 2 In what follows, we write Xr for Xr = e
2π in2 r
r . 0
Lemma 8. Let G be a simple compact Lie group with Coxeter number h, r ≥ 5 a prime, and ρ : G → GL(V ) an irreducible representation of G such that r divides dim V . If there exists g ∈ G such that the spectrum of ρ(g) is Xr , then r ≤ 2h − 5. Proof. Let T be a maximal torus containing g and ( , ) denote the Cartan-Killing form on the character space X∗ (T ) ⊗ R. Let β, α =
2(β, α) , (α, α)
and fix a Weyl chamber. If V has highest weight λ and ρ is the half sum of positive roots, the Weyl dimension formula ([Bo] VIII, §9, Th. 2) asserts dim V =
λ, α + ρ, α , ρ, α
α>0
where the product is taken over all positive roots. Let β denote the highest root. Since r divides dim V , and µ, α ∈ Z for all weights µ and roots α, λ, β + ρ, β ≥ r, or, by [Bo] VI, §1, Prop. 29(c) and [Bo] VI, §1, Prop. 31, λ, β + h − 1 ≥ r. The string of weights λ, λ − β, . . . has length 1 + λ, β ([Bo] VIII, §7, Prop. 3(i)). For any w in the Weyl group, the string w(λ), w(λ) − w(β), . . . has the same length. The Weyl orbit of β consists of all long roots ([Bo] VI, §1, Prop. 11), so the lattice it generates contains the root lattice if G is of type A, D, or E; twice the root lattice if G is of type B, C, or F; and three times the root lattice if G is of type G ([Bo] Planches). Since the difference between weights in an irreducible representation belongs to the root lattice, and since the eigenvalues of ρ(g) are r th roots of unity, not all equal, we conclude that if r ≥ 5, w(β)(g) is a primitive r th root of unity for some w ∈ W . A non-trivial geometric progression in Xr has length ≤ r−1 2 , so 1 + λ, β ≤ Thus, r ≤ 2h − 5.
r −1 . 2
Lemma 9. Under the hypotheses of Lemma 8, if dim V = group. If V is not self-dual, then G is of type An .
r 3 −r 24 ,
then G is a classical
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Proof. The Coxeter numbers of G2 , F4 , E6 , E7 , E8 are 6, 12, 12, 18, 30, respectively ([Bo] Planches). Examination of all primes ≤ 55 [MP] reveals that the only case in 3 −r is the dimension of a representation of a suitable exceptional group is r = 7. which r 24 The group is G2 , and V is the adjoint representation. This case is excluded, however, as the longest string of short roots is 4 > r−1 2 . Thus β(g) = 1 for all short roots β. As the short roots generate the root lattice, g has all eigenvalues equal, contrary to assumption. If V is not self-dual, G cannot be of type Bn or Cn and can only be of type Dn if n is odd and the highest weight λ = a1 λ1 + · · · + an λn satisfies sup(an−1 , an ) > 0 [MP]. As the Weyl dimension formula is monotonic in each ai , dim V ≥ 2n−1 while r ≤ 2h − 5 = 4n − 9. Given the dimension of V , the only possibilities for (n, r) are (5, 11), (7, 13), (7, 17), (7, 19), (9, 19), (9, 23), (11, 31). For r ≤ 31, the longest geometric progression in Xr has length ≤ 4, so h ≥ r − 2 or r ≤ 2n. This leaves only the case (7, 13), and by [MP], the only irreducible 91-dimensional representation of D7 is self-dual. Lemma 10. For n ≥ 11, if G = SU(n) and V is an irreducible representation of G with highest weight λ and dimension < 2 n3 , then λ is one of the following: Highest weight Dimension 0 λ1 , λn−1 λ2 , λn−2 2λ1 , 2λn−1 λ1 + λn−1 λ3 , λn−3 3λ1 , 3λn−1
1 nn n+12 2
n2 − 1 n
n+23 3
Proof. By the monotonicity of the Weyl dimension formula, it is enough to check that the dimension is always greater than 2 n3 in the following cases (and their duals): λk , 4 ≤ k ≤ n − 4; 4λ1 ; 2λ2 ; 2λ3 ; λ1 + λ2 ; λ1 + λ3 ; λ2 + λ3 ; λ1 + λn−2 ; λ1 + λn−3 ; λ2 + λn−2 ; λ2 + λn−3 ; λ3 + λn−3 ; 2λ1 + λn−1 . Lemma 11. Let r ≥ 7 be a prime, G a simple compact Lie group, and ρ : G → GL(V ) an irreducible representation of G such that 1) V is not self dual, 3 −r , 2) dim V = r 24 3) There exists g ∈ G such that the spectrum of ρ(g) is Xr . Then either G = SU(dim V ), and V is the standard representation or its dual; or r = 13, G is a central quotient of SU(14), and V is the exterior square representation or its dual. Proof. By Lemma 8, r ≤ 2h − 5. By Lemma refclass, G is of type An , so n ≥ r3 − r n (r + 3)(r + 1)(r − 1) > . 2 ≥ 24 24 3
r+3 2 ,
so
If n + 1 < 11, r < 17, so there are three cases: r = 7, n ≥ 5, and dim V = 14; r = 11, n ≥ 7, and dim V = 55; and r = 13, n ≥ 8, and dim V = 91. For n ≤ 10, we see there is just one possibility for an irreducible representation of SU(n + 1) of the given
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dimension: r = 11, n = 10, and V is the exterior square of the standard representation of SU(11) or its dual. If n + 1 ≥ 11, by Lemma 10, either V or V ∗ has highest weight in the set λ1 , 2λ1 , 3λ1 , λ2 , λ3 , λ1 + λn−1 . As V is not self-dual, we can exclude λ1 + λn−1 . For r ≥ 7, (r + 1)r(r − 1) (r − 1)(r − 2)(r − 3) < , 24 8
3 −r so m3 ≤ r 24 only when m < r − 1, in which case equality is ruled out since r does m not divide 3 . Applying this when m = n ± 1, we exclude the cases λ3 and 3λ1 . For λ2 and 2λ1 , we seek solutions of m r3 − r = 2 24 in integers m. For any such solution, r|m or r | m − 1, so 12a(ar ± 1) = r 2 − 1, a ∈ N. The discriminant of the quadratic equation for r is (12a 2 )2 + 4 ± 48a, and (12a 2 − 1)2 < (12a 2 )2 + 4 − 48a < (12a 2 )2 < (12a 2 )2 + 4 + 48a < (12a 2 + 1)2 for a ≥ 3. For a = 2, the discriminant is not square for either choice of sign. For a = 1, we get the two solutions (r, m) = (11, 11) and (r, m) = (13, 14). An exhaustive analysis of sets S of r th roots of unity, whose symmetric or exterior squares give X11 or X13 reveals exactly two possibilities: the set 3 9 , ζ13 } {1, ζ13 , ζ13
and its complex conjugate have exterior square X13 .
(1)
Let ρg : Mg → PGL(Vg ) denote the projective unitary representation given by the SO(3)-theory. Let Gg denote the closure of the image. It is a subgroup of PSU(dim Vg ) and therefore a compact Lie group. We will often regard Vg as a projective representation of Gg . Theorem 2. For g = 2, the projective representation ρ associated to the SO(3) theory 2π i at A = ie 4r for r ≥ 5 has dense image. The proof will be carried out in several steps. By [FLW2], we may assume r ≥ 7. Step 1. The Lie group G2 is infinite. Proof. Consider the decomposition of the representation space arising from a curve separating a genus 2 surface into two genus 1 surfaces with boundary. The components are indexed by labels 0, 2, . . . , r − 3, and they are projective representation spaces of M1 × M1 . The representation associated with label 2l is of the form W2l ⊗ W2l , where each tensor factor has dimension r−1−2l . For label r − 5, it has dimension 2. Thus we 2 have a two-dimensional projective unitary representation of M1 = SL2 (Z); the ratio of eigenvalues for a Dehn twist is a primitive r th root of unity. By the classification of finite subgroups of SO(3), this implies the image is infinite, and it follows that the same is true for G2 .
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Step 2. The projective representation V2 is not self-dual. Proof. Equivalently, for any central extension M˜ 2 for which one can lift V2 to a linear representation (also denoted V2 ), the contragredient representation V2∗ is not obtained by tensoring V2 by a central character. We compute the multiplicities of the eigenvalues of a lift to M˜ 2 of a Dehn twist. These are just the dimensions dr,1,2l,2l of a doubly-punctured torus with both labels equal to 2l, and are therefore given by dr,1,2l,2l =
(2l + 1)(r − 2l − 1) . 2
No two of these multiplicities coincide as l ranges over integers ≤ self-dual.
(3) r−3 2 , so V2
cannot be
˜ 2 denote any central extension of G2 for which V2 lifts to a linear repreStep 3. Let G ˜ ◦ denote the identity component of G ˜ 2. sentation (which we also denote (ρ2 , V2 )). Let G 2 ◦ ˜ Then the restriction of V2 to G2 is isotypic. ˜ 2 of a Dehn twist t would Proof. As M2 is generated by Dehn twists, some lift t˜ ∈ G otherwise permute the isotypic components non-trivially. Thus, the eigenvalues of ρ2 (t˜) (which are defined up to multiplication by a common scalar) would contain a coset of a non-trivial group of roots of unity [FLW2] Lemma 1.2. This is impossible, since up to scalars, the spectrum of a Dehn twist is Xr . ˜ such ˜ 2 of G2 as above and any normal subgroup G Step 4. For any central extension G 2 ˜ 2 /G ˜ is trivial, V2 is tensor indecomposable that every homomorphism SL2 (Fr ) → G 2 ˜ -representation. as a G 2 Proof. The restriction of V2 to ˜2 M˜ 1 = {1} × M1 ⊂ M 1 × M1 ⊂ M decomposes as a sum of terms of the form W2l . Let H˜ 1 denote the closure of M˜ 1 in ˜ . The occurrence of W0 as a factor guarGL(V2 ) and H˜ 1 the intersection of H˜ 1 with G 2 antees that SL2 (Fr ) or PSL2 (Fr ) is a quotient of H˜ 1 for which the W0 -factor in question ˜ ˜ is an irreducible representation of degree r−1 2 ; the condition on G2 guarantees that H1 maps onto SL2 (Fr ), so V2 |H˜ has an irreducible factor which is the composition of this 1
quotient map and a degree r−1 2 representation of SL2 (Fr ). Tensor indecomposability now follows from Lemma 7. Step 5. The restriction of V2 to G◦2 is irreducible. ˜ 2 of G2 so that regardProof. Otherwise, by Lemma 3, there exists a central extension G ˜ ing V2 as a G2 -representation, it has a tensor-decomposition. Step 6. The identity component G◦2 is a simple compact Lie group.
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Proof. The identity component K = G◦2 is a connected compact Lie group. As V2 is an irreducible projective representation, the center of K is trivial. Therefore, it is a product of compact simple Lie groups K1 × · · · × Ks , and V2 is a tensor product of unitary projective representations X1 , . . . Xs of the Ki . In other words, K1 × · · · × Ks → PSU(X1 ) × · · · × PSU(Xs ) → PSU(V2 ), where the first inclusion is the product of inclusion Ki → PSU(Xi ) and the second 3 −r for all r, s < r. Consider the composition is the tensor product map. As 2r > r 24 π : G2 → Aut(K) → Out(K). The outer automorphism group is contained in a product of groups of the form Out(Ki )si Ssi , where si ≤ s. The largest proper subgroup of SL2 (Fr ) is the Borel subgroup with index r + 1 > si , so any homomorphism SL2 (Fr ) → Ssi is trivial. Therefore, any homomorphism from SL2 (Fr ) to Out(K) lands in a product of solvable groups, and since SL2 (Fr ) is perfect, that means any such homomorphism is trivial. By Step 4, V2 is tensor indecomposable as a representation of any central extension of the subgroup G2 = ker π. However, g acts by inner automorphisms on K for g ∈ G2 , and since ρ2 is irreducible, this means G2 ⊂ K ⊂ si=1 PSU(Xi ), so s = 1. Step 7. The theorem holds if r = 13. Proof. Applying Lemma 11 to the universal cover of G◦2 , G◦2 is all of PSU(dim V2 ), so the same is true for G2 . Step 8. The theorem holds if r = 13. Proof. For r = 13, we must consider the possibility that the universal covering group of G◦2 is SU(14), V2 is its alternating square, and a Dehn twist has exactly four different eigenvalues λi in SU(14) given up to a common scalar multiple by (1) or its complex conjugate. By (3), the eigenvalues of a Dehn twist have multiplicities 6, 15, 20, 21, 18, 11 in PSU(dim V2 ), and each one arises uniquely as a product of distinct eigenvalues λi . Therefore, some λi must have multiplicity 11, but this is impossible since only one of the eigenvalues in V2 has multiplicity divisible by 11. Therefore, G◦2 = PSU(dim V2 ) also for r = 13. 5. Case g ≥ 3 Theorem 3. For all r ≥ 5 and all g ≥ 2, ρg (Mg ) is dense in PSU(dim Vg ). For r = 5, this is already known [FLW2] Theorem 6.2. We therefore assume from now on that r ≥ 7. We begin with a dimension estimate. Lemma 12. For r ≥ 7 and g ≥ 2, dim Vg+1 < except when r = 7 and g = 2.
dim Vg 2
(2)
Density of the SO(3) TQFT Representation of Mapping Class Groups
Proof. As αk > 1 for all k, if g ≥ 3, dim Vg+1 =
g αk
≤
k
655
3(g−1)/2 αk
< dim
3/2
Vg
k
dim Vg < . 2
For g = 2, we compute (r + 5)(r + 3)(r + 1)r(r − 1)(r − 8) dim V2 − dim V3 = , 2 5760 and this quantity is obviously positive when r > 7.
Proof of Theorem 3. The proof is very similar to that of Theorem 2.
Step 9. The Lie group Gg is infinite. Proof. Consider the decomposition of the representation space arising from a curve separating a genus g surface into two pieces, one of genus 1 and one of genus g − 1. Restricting to Mg−1 × M1 , we obtain a decomposition r−3
Vg =
2
Xg−1,2l ⊗ W2l ,
l=0
where Xg−1,2l denotes the projective representation space of Mg−1 associated to a surface of genus g − 1 with a single boundary component labeled 2l. Now we proceed as in Step 1 of Theorem 2. Step 10. The projective representation Vg is not self-dual. Proof. If Vg is self-dual, its restriction to Mg−1 × M1 decomposes into self-dual projective representations and mutually dual pairs of projective representations. We use induction on g, the base case being Step 2 of Theorem 2. By the induction hypothesis, Xg−1,0 ⊗ W0 = Vg−1 ⊗ W0 is not self-dual. Neither can it be dual to any other factor since W0 = V1 is irreducible and the other representations W2l have lower dimension. ˜ g denote any central extension of Gg for which Vg lifts to a linear repreStep 11. Let G ˜ ◦g denote the identity component of G ˜ g . Then the restriction of Vg to G ˜ ◦g sentation. Let G is isotypic. Proof. Identical to the proof of Step 3 of Theorem 2.
˜ g of Gg as above and any normal subgroup, Vg is Step 12. For any central extension G ˜ g -representation. tensor indecomposable as a G Proof. The restriction of Vg to M˜ 1 = {1} × M1 ⊂ Mg−1 × M1 ⊂ M˜ g decomposes as a sum of terms of the form W2l . Now we proceed as in Step 4 of Theorem 2.
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Step 13. The restriction of Vg to G◦g is irreducible. Proof. Identical to the proof of Step 5 of Theorem 2.
Step 14. The identity component G◦g is a simple compact Lie group. Proof. Let K = G◦g . As in Step 6 of Theorem 2, K is a product of compact simple Lie groups Ki , and Vg is the tensor product of unitary projective representations Xi of the Ki . Thus, K1 × · · · × Ks → PSU(X1 ) × · · · × PSU(Xs ) → PSU(Vg ). The conjugation action of G2 on K must act transitively on the factors, since any decomposition into orbits gives a tensor decomposition of V . Therefore, the Ki are mutually isomorphic, and their representations Xi are equivalent up to composition with automorphisms of Xi ; in particular, their degrees are all the same. Now, the closure of Mg−1 ⊂ Mg−1 × M1 in K maps onto Gg−1 since Vg−1 = Xg−1,0 is a summand of the restriction of Vg to Mg−1 . Thus some factor Ki maps onto Gg−1 . By the induction hypothesis, 2 − 1, dim Ki ≥ dim Gg−1 = dim Vg−1
so any non-trivial representation of Ki has degree ≥ dim Vg−1 . Therefore, dim Vg ≥ s . The inequality (2) then implies s = 1 except possibly when g = 3 and r = 7, dim Vg−1 2 , so again s = 1. in which case, dim Vg = 98 < 142 = dim Vg−1 Step 15. For all g ≥ 3 and r ≥ 7, Gg = PSU(dim Vg ). Proof. We use induction, the base case being Theorem 2. By the induction hypothesis and (2), rank G◦g ≥ rank Gg−1 ≥ dim Vg−1 − 1 > 2 dim Vg + 1/4 − 1/2. (4) By Step 2, Vg is not self-dual, so by [MP], Gg is of type An , Dn (n odd), or E6 . The case E6 is ruled out since dim Vg−1 > 7 in all cases g ≥ 3. The minimal dimension for a representation of Dn which is not self-dual is 2n−1 > n+1 for n > 4. This leaves 2 n+1 the case An , where the inequality dim Vg < 2 implies dim Vg < 2 n+1 for n > 2. 3 Lemma 10 gives the list of possibilities. For n > 7, the possible highest weights are λ1 , λn , λ2 , λn−1 , 2λ1 , 2λn , and λ1 + λn , the last being ruled out as Vg is not self-dual. Up to duality, then, Vg is either the standard representation, its exterior square, or its symmetric square, and both are ruled out by (4). 6. An Application Besides determining the representations of the mapping class groups, a TQFT also determines invariants of oriented closed 3-manifolds [RT]. To describe the SO(3) invariant of 3-manifolds, we introduce the following notations: let di and θi be the quantum dimension and the twist of the label i, and D the global dimension of the modular ten sor category defined in Sect. 1; then we define p± = i∈L θi±1 di2 , and ω0 to be the formal sum i∈L dDi ·i . If a 3-manifold M 3 is represented by a framed link L in S 3 , then the SO(3) 3-manifold invariant of M 3 is τ (M 3 ) = D1 · ω0 ∗L · ( pD− )σ (L) , where
Density of the SO(3) TQFT Representation of Mapping Class Groups
657
σ (L) is the signature of the linking matrix of L, and ω0 ∗L is the link invariant of L where each component of L is labeled by ω0 [T]. Note that in this normalization, τ (S 1 × S 2 ) = 1, τ (S 3 ) = D1 . The invariant τ is multiplicative for disjoint unions, but for connected sums τ (M1 #M2 ) = D · τ (M1 )τ (M2 ). Recall that a TQFT 3-manifold invariant is only defined for extended oriented closed 3-manifolds, but as pointed out in [A] there is a preferred framing for each oriented closed 3-manifold; therefore, the formula above should be thought of as for an extended 3-manifold with the preferred framing determined by the framed link L. The same 3manifold invariant can also be defined using the representations of the mapping class groups. The subtlety in framing is reflected in the fact that the TQFT representations of the mapping class groups are only projective representations. It is known that pD− is π ic
a root of unity of finite order [BK], so we can write pD− = e 4 for a rational number c, which is called the central charge of the TQFT (well-defined modulo 8). Framing π ic changes lead to powers of κ = e 4 . So up to powers of κ, the same SO(3) invariant of 3-manifolds can be obtained as follows: suppose an oriented 3-manifold M 3 is given by gluing together two genus=g oriented handlebodies Hg by a self-diffeomorphism f of g = ∂Hg (note that the handlebody Hg determines a vector v0 in the TQFT vector space Vr (g ) of g up to a power of κ); then the SO(3) 3-manifold invariant τ (M 3 ) is, up to a power of κ, the inner product of v0 with ρSO(3) (f )(v0 ). Theorem 3 has a direct corollary concerning the Witten-Reshetikhin-Turaev SO(3) invariants of 3-manifolds. Theorem 4. The set of the norms of the SO(3) invariants at A = ie 3-manifolds is dense in [0, ∞) for primes r ≥ 5.
2π i 4r
of all connected
Proof. Given any complex number z, note that D > 1 so we can find a g > 1 so that z z = D g−1 satisfies |z | < 1. Then arrange z as the (1, 1) entry of a unitary matrix Uz , and the vector v0 determined by Hg as the first basis vector of an orthonormal basis of Vr (). By Theorem 3, Uz can be approximated by a sequence of unitary matrices associated to diffeomorphisms fi of g up to a phase. Each fi determines a 3-manifold Mfi by gluing two copies of Hg . It follows that the 3-manifold invariant of Mfi is the (1,1)-entry of ρSO(3) (fi ) up to a phase, hence approximates z up to a phase. By connecting sums Mfi with (g − 1) copies of S 1 × S 2 , we approximate z using the invariants of the 3-manifolds Mfi #(g − 1)(S 1 × S 2 ). Remark. (1) A similar density result of the SU(2) invariants for r = 1 mod 4 can be deduced from the SO(3) case and the decomposition formula. (2) This result does not follow from Funar’s result [F]. The infinite mapping classes are in the image of the braid groups, and can be extended to the handlebody groups of Hg . Therefore, all resulting 3-manifolds have the same invariant up to powers of κ. Finally, we make the following: Conjecture. The set of the SO(3) invariants at A = ie is dense in the complex plane for primes r ≥ 5.
2π i 4r
of all connected 3-manifolds
References [A] [At]
Atiyah, M.: On framings of 3-manifolds. Topology 29, 1–7 (1990) Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A.Wilson, R.A.: Atlas of Finite Groups. Oxford: Clarendon Press, 1985
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Bourbaki, N.: Groupes et alg`ebres de Lie. Paris: Hermann, Paris, 1968, Chapter 4–6, 1975, Chapter 7–8 [BK] Bakalov, B., Kirillov, A.: Lectures on tensor categories and modular functors. University Lecture Series Vol. 21, Providence, RI: AMS, 2000 [EM] Eilenberg, S., Mac Lane, S.: Cohomology theory in abstract groups. II. Group extensions with a non-Abelian kernel. Ann. Math. 48, 326–341 (1947) [F] Funar, L.: On the TQFT representations of the mapping class groups. Pacific J. Math. 188(2), 251–274 (1999) [FKLW] Freedman, M., Kitaev, A., Larsen, M., Wang, Z.: Topological quantum computation. Bull. AMS 40, 31–38 (2003) [FLW1] Freedman, M., Larsen, M., Wang, Z.: A modular functor which is universal for quantum computation. Commun. Math. Phys. 227, 605–622 (2002) [FLW2] Freedman, M., Larsen, M., Wang, Z.: The two-eigenvalue problem and density of Jones representation of braid groups. Commun. Math. Phys. 228, 177–199 (2002) [FK] Frohman, C., Kania-Bartoszynska, J.: SO(3) topological quantum field theory, Commun. Anal. Geom. 4(4), 589–679 (1996) [FNWW] Freedman, M., Nayak, C., Walker, K., Wang, Z.: Picture TQFTs, in preparation [G] Gilmer, P.: On the Witten-Reshetikhin-Turaev representations of mapping class groups. Proc. Amer. Math. Soc. 127(8), 2483–2488 (1999) [Ge] Gerardin, P.: Weil representations associated to finite fields. J. Algebra 46(1), 54–101 (1977) [J] Jeffrey L.: Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation. Commun. Math. Phys. 147(3), 563–604 (1992) [KL] Kauffman, L., Lins, S.: Temperley-Lieb recoupling theory and invariants of 3-manifolds. Ann. Math. Studies, Vol. 134, Princeton, NJ: Princeton University Press, 1994 [MP] Mckay, W., Patera, J.: Tables of dimensions, indices, and branching rules for representations of simple Lie algebras, Lecture Notes in Pure and Applied Math., Vol. 69 [R] Roberts, J.: Irreducibility of some quantum representations of mapping class groups. J. Knot Theory Ramifications 10(5), 763–767 (2001) [RT] Reshetikhin, N., Turaev, V.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103, 547–598 (1991) [Sp] Springer, T.A.: Characters of Special Groups. In: Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Mathematics 131, Berlin:Springer-Verlag, 1970 [T] Turaev, V.: Quantum invariants of knots and 3-manifolds. Berlin: de Gruyter Studies in Math., Vol 18, 1994 [Wi] Witten, E.:Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989) [Wr1] Wright, G.: The Reshetikhin-Turaev representation of the mapping class group. J. Knot Theory Ramifications 3(4), 547–574 (1994) [Wr2] Wright, G.: The Reshetikhin-Turaev representation of the mapping class group at the sixth root of unity. J. Knot Theory Ramifications 5(5), 721–739 (1996) Communicated by M.B Ruskai
Commun. Math. Phys. 260, 659–671 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1426-2
Communications in
Mathematical Physics
A Quantum Version of Sanov’s Theorem Igor Bjelakovi´c1 , Jean-Dominique Deuschel1 , Tyll Krüger1,2 , Ruedi Seiler1 , Rainer Siegmund-Schultze1,3 , Arleta Szkoła1,4 1
Institut für Mathematik MA 7-2, Technische Universität Berlin, Fakultät II - Mathematik und Naturwissenschaften, Straße des 17. Juni 136, 10623 Berlin, Germany. E-mail: {igor, deuschel, tkrueger, seiler, siegmund, szkola}@math.tu-berlin.de 2 Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33619 Bielefeld, Germany 3 Institut für Mathematik, Technische Universität Ilmenau, 98684 Ilmenau, Germany 4 Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany Received: 22 December 2004 / Accepted: 31 March 2005 Published online: 20 September 2005 – © Springer-Verlag 2005
Abstract: We present a quantum version of Sanov’s theorem focussing on a hypothesis testing aspect of the theorem: There exists a sequence of typical subspaces for a given set of stationary quantum product states asymptotically separating them from another fixed stationary product state. Analogously to the classical case, the separating rate on a logarithmic scale is equal to the infimum of the quantum relative entropy with respect to the quantum reference state over the set . While in the classical case the separating subsets can be chosen universally, in the sense that they depend only on the chosen set of i.i.d. processes, in the quantum case the choice of the separating subspaces depends additionally on the reference state. 1. Introduction In this article we present a natural quantum version of the classical Sanov’s theorem as part of our attempt to explore basic concepts and results at the interface of classical information theory and stochastics from the point of view of quantum information theory. Among those classical results a crucial role is played by the Shannon-McMillanBreiman theorem (SMB theorem). It clarifies the concept of typical subsets and provides the rigorous background for asymptotically optimal lossless data compression. It says that a long message from an ergodic data source belongs most likely to a typical subset (of the generally very much larger set of all possible messages). The cardinality within the sequence of these typical sets grows with the length n of the message at an exponential rate given by the Shannon entropy rate of the data source. As a general rule, when passing to the quantum situation, the notion of a typical subset is replaced by a typical subspace in the corresponding Hilbert space describing the pure n-block states of the quantum data source. The dimension of the subspace is growing exponentially fast at a rate given by the von Neumann entropy rate as n goes to infinity (cf. [11] in the i.i.d. situation, [1, 2] in the general ergodic case).
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We know from the classical situation that typical subsets have even more striking properties, when chosen in the right way: For a given alphabet A and a given rate there is a universal sequence of typical subsets growing at the given rate for all ergodic sources with entropy rate smaller than the given rate (this result has been generalized to the quantum context by Kaltchenko and Yang [12]). Moreover, for any ergodic data source P we can find a sequence of typical subsets growing at the rate given by the entropy and at the same time separating it exponentially well from any i.i.d. (reference) data source Q in the sense that the Q-probability of the entire P -typical subset goes to zero at an exponential rate given by the relative entropy rate h(P , Q). Furthermore, the relative entropy is the best achievable (optimal) separation rate. This assertion which gives an operational interpretation of the relative entropy is Stein’s lemma. We mention that the i.i.d. condition concerning the reference source cannot be weakened too much, since there are examples where even the relative entropy has no asymptotic rate, though the reference source is very well mixing (B-process, cf. [16]). A quantum generalization of this result can be found in [13] for the case that both sources are i.i.d., and in [5] for the case of a general ergodic quantum information source. This result was mainly inspired by [10], where complete ergodicity was assumed and optimality was still left open. From the viewpoint of information theory or statistical hypothesis testing the essential assertion of Sanov’s theorem is that it represents a universal version of Stein’s lemma by saying that for a set of i.i.d. sources there exists a common choice of the typical set such that the probability with respect to the i.i.d. reference source Q goes to zero at a rate given by inf P ∈ h(P , Q). Originally Sanov’s theorem is of course a result on large deviations of empirical distributions (cf. [15, 8]). It is the information-theoretical viewpoint taken here which suggests to look at it as a large deviation principle for typical subsets. With the main topic of this paper being a quantum theorem of Sanov type, it is especially appealing to shift the focus from empirical distributions to typical subspaces, since the notion of an individual quantum message string is at least problematic, and as will be seen by an example, a reasonable attempt to define something like quantum empirical distributions via partial traces leads to a separation rate worse than the relative entropy rate (see the last section). Another aspect of the classical Sanov result has to be modified for the quantum situation: The typical subspace will no longer be universal for all i.i.d. reference sources, but has to be chosen to depend on the reference source. So only ‘one half’ of universality is maintained when passing to quantum sources, namely that which refers to the set . This again will be demonstrated by an example in the last section. The basic mechanism behind this no go result is - heuristically speaking: In the quantum setting even pure states cannot be distinguished with certainty, while classical letters can. In our forthcoming paper [4] we extend the results given here to the case where only stationarity is assumed for the states in . 2. A Quantum Version of Sanov’s Theorem Let A be a finite set with cardinality #A = d. By P(A) we denote the set of probability distributions on A. The relative entropy H (P , Q) of a probability distribution P ∈ P(A) with respect to a distribution Q is defined as usual: a∈A P (a)(log P (a) − log Q(a)), if P Q H (P , Q) := (1) ∞, otherwise,
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where log denotes the base 2 logarithm. For the base e logarithm we use the notation ln. The function H (·, Q) is continuous on P(A), if the reference distribution Q has full support A. Otherwise it is lower semi-continuous. The relative entropy distance from the reference distribution Q to a subset ∈ P(A) is given by: H (, Q) := inf H (P , Q). P ∈
(2)
Our starting point is the classical Sanov’s theorem formulated from the point of view of hypothesis testing: Theorem 1 (Sanov’s Theorem). Let Q ∈ P(A) and ⊆ P(A). There exists a sequence {Mn }n∈N of subsets Mn ⊆ An with lim P n (Mn ) = 1,
n→∞
∀P ∈ ,
(Typicality)
(3)
such that 1 log Qn (Mn ) = −H (, Q). (Separation rate) n→∞ n lim
(4)
Moreover, for each sequence of sets {M˜ n } fulfilling (3) we have lim inf n→∞
1 log Qn (M˜ n ) ≥ −H (, Q), n
such that H (, Q) is the best achievable separation rate. We emphasize that in the above formulation we omitted the assertion that the sets Mn can be chosen independently from the reference distribution Q. However, as will be shown in the last section, in the quantum case this universality feature is not valid any longer and Theorem 1 is the strongest version that has a quantum analogue. It is an immediate consequence of Lemma 1 and is related to the usual formulation of Sanov’s theorem([15], see also Theorem 3.2.21 in [8]) in terms of empirical measures Px n := n1 ni=1 δxi for sequences x n := {x1 , . . . , xn } as follows: By the strong law of large numbers, the sequence of empirical distributions {Px n } formed along an i.i.d. sequence {x1 , x2 , . . . } of letters distributed according to a probability measure P ∈ P(A) tends to P almost surely. Thus for any neighbourhood U of P we have limn→∞ P n ({x n : Px n ∈ U }) = 1; hence the sequence of sets {x n : Px n ∈ U } is typical for {P n }n∈N . If is an open set, then we may choose U as and {x n : Px n ∈ U } is universally typical for all P n , P ∈ . Now Sanov’s theorem in its traditional form says that − n1 log Qn ({x n : Px n ∈ }) → H (, Q). So the (explicitely specified) typical sets separate Qn exponentially fast from all P n , P ∈ with the given order H (, Q). Passing to the quantum setting we substitute the set A by a C ∗ -algebra A with n dimension dim A < ∞ and the cartesian product A = ni=1 A by the tensor prod uct A(n) := ni=1 A. Note that any finite dimensional C ∗ -algebra A is ∗-isomorphic to a direct sum of full matrix algebras (cf. [17]), i.e. A kj =1 Mnj (C) for suitable positive integers n1 , . . . , nk . We denote by S(A) the set of quantum states on the algebra of observables A, i.e. S(A) is the set of positive functionals ϕ on A fullfilling the normalisation condition ϕ(1) = 1. It is well known (cf. [14]) that each state ϕ ∈ S(A) can be represented by a unique density operator Dϕ ∈ A, i.e. ϕ(a) = tr(Dϕ a) for all a ∈ A, where tr denotes the image of the canonical trace on the algebra kj =1 Mnj (C)
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under the ∗-isomorphism mentioned a few lines above. For ϕ ∈ S(A) we mean by ϕ ⊗n a product state on A(n) . Note that for corresponding density operators Dϕ ⊗n = Dϕ⊗n holds. The quantum relative entropy S(ψ, ϕ) of the state ψ ∈ S(A) with respect to the reference state ϕ ∈ S(A) is defined by: trA Dψ (log Dψ − log Dϕ ), if supp(ψ) ≤ supp(ϕ) S(ψ, ϕ) := (5) ∞, otherwise. Observe that in the case of a commutative C ∗ -algebra A the quantum relative entropy S coincides with the classical relative entropy H defined in (1), where the probabilities are defined as the expectations of minimal projectors in A . The functional S(·, ϕ) is continuous on S(A) only if the reference state ϕ is faithful, i.e. suppϕ = 1A , otherwise it is lower semi-continuous. The relative entropy distance from the reference state ϕ to a subset ⊆ S(A) is given by: (6)
S(, ϕ) := inf S(ψ, ϕ). ψ∈
Now we are in the position to state our main result: Theorem 2 (Quantum Sanov’s Theorem). Let ϕ ∈ S(A) and ⊆ S(A). There exists a sequence {pn }n∈N of orthogonal projections pn ∈ A(n) such that lim ψ ⊗n (pn ) = 1,
n→∞
∀ψ ∈ ,
(Typicality)
(7)
and lim
n→∞
1 log ϕ ⊗n (pn ) = − inf S(ψ, ϕ). (Separation rate) ψ∈ n
Moreover, for each sequence of projections {p˜ n } satisfying Eq. (7) we have lim inf n→∞
1 log ϕ ⊗n (p˜ n ) ≥ − inf S(ψ, ϕ), ψ∈ n
such that S(, ϕ) is the best achievable separation rate. The proof of Theorem 2 will be based to a large extent on the following classical lemma, which is a stronger version of Theorem 1. Lemma 1. Let Q ∈ P(A) and ⊆ P(A). For each sequence {εn }n∈N satisfying εn 0 and log(n+1) → 0 there exists a sequence {Mn }n∈N of subsets Mn ∈ An such that for nεn2 each P ∈ there is an N (P ) ∈ N with P n (Mn ) ≥ 1 − (n + 1)#A · 2−nbεn , 2
∀n ≥ N (P ),
(8)
where b is a positive number. Moreover we have: 1. lim inf n→∞ n1 log Qn (Mn ) ≥ −H (, Q). 2. Qn (Mn ) ≤ (n + 1)#A · 2−nIn , ∀n ∈ N, where In ≥ 0 and for all n ∈ N fulfilling εn ≤ 21 , 0 ≤ H (, Q) − In ≤ log(#A)εn − εn log εn − εn log Qmin , holds with Qmin := min{Q(a) : Q(a) > 0, a ∈ A}.
(9)
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Proof. Due to the classical Stein’s lemma any sequence of subsets {Mn }n∈N , which has an asymptotically non-vanishing measure with respect to the product distributions P n satisfies: 1 lim inf log Qn (Mn ) ≥ −H (P , Q), (10) n→∞ n where Q ∈ P(A) is the reference distribution. Then the lower bound (10) implies the first item of Lemma 1. We partition the set into the set 1 consisting of probability distributions which are absolutely continuous w.r.t. Q and its complement 2 within , i.e. 1 := {P ∈ : H (P , Q) < ∞}, Observe that
H (, Q) =
and
2 := c1 ∩ .
H (1 , Q), if 1 = ∅ ∞, otherwise
(11)
holds. We will treat these two sets separately. It is obvious that we can perfectly distinguish the distributions in 2 from Q, we just have to set M2,n := {x n ∈ An : Qn (x n ) = 0 and P n (x n ) > 0 for some P ∈ 2 }.
(12)
Then we have for all n ∈ N, Qn (M2,n ) = 0.
(13)
Moreover we have for each P ∈ 2 and n ∈ N, P n (M2,n ) = 1 − qPn → 1
(n → ∞),
(14)
where qP := P (A+ ), with A+ := {a ∈ A : Q(a) > 0}.
(15)
Observe that the speed of convergence in (14) is exponential. In treating the set 1 we may consider the restricted alphabet A+ , defined in (15), only. Note that H (·, Q) is continuous as a functional on P(A+ ). Choose a sequence εn 0 with log(n+1) → 0 and define the following decreasing family of sets: nε2 n
n := {R ∈ P(A+ ) : R − P 1 ≤ εn for at least one P ∈ 1 }. Observe that n 1 , the closure of 1 . Moreover we set M1,n := {x n ∈ An+ : Px n ∈ n },
(16)
where Px n denotes the empirical distribution or type of the sequence x n . Now, by type counting methods (cf. [7], Sect. 12.1) and Pinsker’s inequality H (P1 , P2 ) ≥ 2 ln1 2 P1 − P2 21 we arrive at c P n (M1,n ) ≤ (n + 1)#A+ 2−nbεn → 0 2
(n → ∞),
(17)
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for each P ∈ 1 where b is a positive number and Mnc denotes the complement of the set Mn . The upper bounds with respect to the distribution Q are a consequence of type counting methods together with the continuity of the functional H (·, Q) on 1 combined with (11) and the fact that n 1 : Qn (M1,n ) ≤ (n + 1)#A+ 2−nH (n ,Q)
(by type counting, cf. [7], Sect. 12.1). (18)
We set In := H (n , Q). Observe that the sequence (In )n∈N is increasing and In ≤ minP ∈1 H (P , Q), for all n ∈ N, since n 1 . Next, observe that n ⊆ {R ∈ P(A+ ) : R − P 1 ≤ εn
for at least one P ∈ 1 }.
(19)
Let Rn ∈ n be such that H (Rn , Q) = minR∈n H (R, Q). By continuity we have H (Rn , Q) = In . According to (19) for each n ∈ N there is a distribution Pn ∈ 1 such that Rn − Pn 1 ≤ εn . Using the inequality |H (P ) − H (R)| ≤ log(#A)P − R1 + η(P − R1 ) valid for distributions P , R with P − R1 ≤ 21 , where η(t) := −t log t, and Qmin := min{Q(a) : a ∈ A+ }, we obtain finally 0 ≤ H (, Q) − In = H (1 , Q) − In = min H (P , Q) − In
(by (11)) (by continuity)
P ∈1
≤ H (Pn , Q) − H (Rn , Q) = H (Rn ) − H (Pn ) + (Rn (a) − Pn (a)) log Q(a) a∈A+
≤ log(#A+ )Pn − Rn 1 + η(Pn − Rn 1 ) −Pn − Rn 1 log Qmin ≤ log(#A)εn − εn log εn − εn log Qmin . Now, setting Mn := M1,n ∪ M2,n , we see by (13) and (18) that for all n ∈ N we have Qn (Mn ) ≤ Qn (M1,n ) + Qn (M2,n ) ≤ (n + 1)#A 2−nIn . Moreover for each P ∈ we may infer from (14) and (17) that for all sufficiently large n ∈ N, P n (Mn ) ≥ 1 − (n + 1)#A 2−nbεn 2
holds.
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3. Proof of the Quantum Sanov’s Theorem Before we prove the Quantum Sanov’s Theorem we cite the relevant known results. We define the maximal separating exponent βε,n (ψ ⊗n , ϕ ⊗n ) := min{log ϕ ⊗n (q) : q ∈ A(n) projection, ψ ⊗n (q) ≥ 1 − ε}. Proposition 1. Let ψ, ϕ ∈ S(A) with the relative entropy S(ψ, ϕ). Then for every ε ∈ (0, 1), lim
n→∞
1 βε,n (ψ ⊗n , ϕ ⊗n ) = −S(ψ, ϕ). n
(20)
The assertion of Proposition 1 was shown by Ogawa and Nagaoka in [13]. A different proof based on the approach of Hiai and Petz in [10] was given in [6]. Proof of Theorem 2. 1. Proof of the lower bound . Due to Proposition 1 any sequence of projections {pn }n∈N , which has an asymptotically non-vanishing expectation value with respect to the stationary product state {ψ ⊗n }n∈N satisfies: lim inf n→∞
1 log ϕ ⊗n (pn ) ≥ −S(ψ, ϕ), n
(21)
where ϕ ∈ S(A) is a fixed reference state. The lower bound (21) implies the lower bound lim inf n→∞
1 log ϕ ⊗n (pn ) ≥ −S(, ϕ) n
for any sequence {pn }n∈N of orthogonal projections pn ∈ A(n) satisfying condition (7) in Theorem 2. 2. Proof of the upper bound . To obtain the upper bound lim sup n→∞
1 log ϕ ⊗n (pn ) ≤ −S(, ϕ), n
where ϕ is a fixed reference state, it is obviously sufficient to show that to each positive δ there exists a sequence pn such that lim ψ ⊗n (pn ) = 1,
n→∞
∀ψ ∈ ,
(22)
and lim sup n→∞
1 log ϕ ⊗n (pn ) ≤ −S(, ϕ) + δ n
is fulfilled for sufficiently large n. To show this we will apply the classical result, Lemma 1, to states restricted to appropriate abelian subalgebras approximating the quasilocal algebra A∞ .
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Consider the spectral decomposition of the density operator Dϕ : Dϕ =
d
λ i ei ,
i=1
where λi are the eigen-values and ei are the corresponding spectral projections. This leads to a decomposition of Dϕ ⊗l = Dϕ⊗l : d l l Dϕ⊗l = λij eij , i1 ,...,il =1
j =1
j =1
which can be rewritten as Dϕ⊗l
=
d
l1 ,...,ld :
i li =l
λlii
el1 ,...,ld ,
i=1
where
el1 ,...,ld :=
l
eij ,
(i1 ,...,il )∈Il1 ...ld j =1
and Il1 ...ld := {(i1 , . . . , il ) : #{j : ij = k} = lk for k ∈ [1, d]}. Let ψ be a state on A and l ∈ N. We denote by Dl,ψ the abelian subalgebra of (l) A generated by {el1 ...ld }l1 ...ld ∪ {el1 ...ld Dψ⊗l el1 ...ld }l1 ...ld . As a finite-dimensional abelian algebra, it has a representation Dl,ψ =
dl
C · fl,i ,
i=1 l is the set of mutually orthogonal minimal projections in Dl,ψ and dl where {fl,i }di=1 denotes the dimension of Dl,ψ . Hiai and Petz [10] have shown that
S(ψ ⊗l , ϕ ⊗l ) = S(ψ ⊗l Dl,ψ , ϕ ⊗l Dl,ψ ) + S(ψ ⊗l ◦ El ) − S(ψ ⊗l ),
(23)
where ψ D denotes the restriction of a state ψ ∈ S(A) to a subalgebra D ⊆ A and El is the conditional expectation with respect to the canonical trace in A(l) : el1 ...ld A(l) el1 ...ld , El : A(l) −→
l1 ...ld :
El (a) :=
l1 ...ld :
i li =l
el1 ...ld ael1 ...ld .
i li =l
Observe that S(ψ ⊗l ◦ El ) − S(ψ ⊗l ) ≤ d log(l + 1),
(cf. [10, 6])
(24)
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which gives the lower bound S(ψ ⊗l Dl,ψ , ϕ ⊗l Dl,ψ ) ≥ S(ψ ⊗l , ϕ ⊗l ) − d log(l + 1)
(25)
implying 1 S(ψ ⊗l Dl,ψ , ϕ ⊗l Dl,ψ ) = S(ψ, ϕ). l→∞ l lim
Next we consider a maximal abelian refinement Bl,ψ of Dl,ψ in the sense of the algebra A(l) : dl ,al,j
Bl,ψ :=
C · gl,j,k ,
j,k=1
where gl,j,k are one-dimensional projections in the algebra A(l) such that fl,j = al,j k=1 C · gl,j,k . This means that Bl,ψ ⊇ Dl,ψ . By monotonicity of the relative entropy and by the estimate (25) we obtain S(ψ ⊗l Bl,ψ , ϕ ⊗l Bl,ψ ) ≥ S(ψ ⊗l Dl,ψ , ϕ ⊗l Dl,ψ ) ≥ l(S(ψ, ϕ) − ηl ),
(26)
where we used the abbreviation ηl := d log(l+1) in the last line. l Due to the Gelfand isomorphism and the Riesz representation theorem the restricted states ψ ⊗l Bl,ψ and ϕ ⊗l Bl,ψ can be identified with probability measures P and Q on the compact maximal ideal space Bl,ψ corresponding to the d l -dimensional abelian algebra Bl,ψ . The relative entropy of P with respect to Q is determined by: H (P , Q) = S(ψ ⊗l Bl,ψ , ϕ ⊗l Bl,ψ ) ≥ l · (S(ψ, ϕ) − ηl ). Similarly, the states ψ ⊗nl Bl,ψ and ϕ ⊗nl Bl,ψ correspond to the product measures n . P n and Qn on the product space Bl,ψ We define (n)
(n)
Sl := inf{S(ψ ⊗l Bl,ψ , ϕ ⊗l Bl,ψ ) : ψ ∈ }, and fix an ψ0 ∈ . For any ψ ∈ and each l ∈ N there exists a unitary operator Uψ ∈ A(l) which transforms the minimal projections spanning Bl,ψ into the minimal projections of Bl,ψ0 and which leaves the spectral subspaces of Dϕ ⊗l invariant. Let us denote by Ul (, ϕ) the set of unitaries having these properties. To each ψ ∈ denote by ψ˜ (l) the state on A(l) with density operator Uψ Dψ⊗l Uψ∗ . Then we have S(ψ ⊗l Bl,ψ , ϕ ⊗l Bl,ψ ) = S(ψ˜ (l) Bl,ψ0 , ϕ ⊗l Bl,ψ0 ). Let l be the set of probability measures on Bl,ψ0 corresponding to all ψ˜ (l) Bl,ψ0 , where ψ ∈ . Further let the measure Q on Bl,ψ0 correspond to the restricted reference state ϕ ⊗l Bl,ψ0 . Then H (l , Q) ≥ Sl ≥ l · (S(, ϕ) − ηl ),
(27)
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where the second inequality follows from (26). Due to Lemma 1 there exists a sequence n {Mn }n∈N of subsets Mn ∈ Bl,ψ (cf. (16)) such that 0 lim P n (Mn ) = 1,
∀P ∈ l ,
n→∞
(28)
and for every n ∈ N, Qn (Mn ) ≤ (n + 1)d 2−nIn (l) , l
where In (l) H (l , Q) for n → ∞. Moreover, we know that In (l) ≥ H (l , Q) − εn (log d l − log εn − log Qmin (l)), where Qmin (l) := min{Q(a) : a ∈ Bl,ψ0 }. We intro(l) duce the abbreviation n := εn (log d l − log εn − log Qmin (l)). We get 1 d l log(n + 1) d l log(n + 1) log Qn (Mn ) ≤ − In ≤ − (H (l , Q)− (l) n ). (29) n n n ⊗n n there corresponds a projection pln in Bl,ψ ⊆ A(nl) . For an arbiTo each Mn ∈ Bl,ψ 0 0 trary m ∈ N such that m = nl + r ∈ N with r ∈ {0, . . . , l − 1} we define a projection pm ∈ A(m) by
pm := pnl ⊗ 1[nl+1,nl+r] , where 1[nl+1,nl+r] denotes the identity in the local algebra A[nl+1,nl+r] . We have ψ˜ (l)⊗n (pnl ) = P n (Mn ),
∀ψ ∈
and 1 1 1 log ϕ ⊗m (pm ) ≤ log ϕ ⊗nl (pnl ) = log Qn (Mn ). m nl nl Using (28), (29) and (27) we conclude lim ψ ⊗nl (Uψ∗⊗n pnl Uψ⊗n ) = 1,
n→∞
∀ψ ∈
(30)
and 1 1 log ϕ ⊗m (pm ) = log Qn (Mn ) m nl d l log(n + 1) 1 ≤ − In (l) nl l d l log(n + 1) 1 ≤ − (H (l , Q)− (l) n ) nl l (l) n d l log(n + 1) − S(, ϕ) + ηl + . ≤ nl l For fixed l ∈ N we construct for each n ∈ N the projection: p nl := U ∗⊗n pnl U ⊗n . U ∈Ul (,ϕ)
(31)
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For an arbitrary number m = nl + r, r ∈ {0, . . . , l − 1}, we define p m := p nl ⊗ 1[nl+1,nl+r] . It follows for arbitrary ψ ∈ and each m = nl + r ∈ N: ψ ⊗m (p m ) = ψ ⊗nl (pnl ) ≥ ψ ⊗nl (Uψ∗⊗n pnl Uψ⊗n ).
(32)
Using the estimate (30) we obtain the general statement: lim ψ ⊗m (pm ) = 1,
∀ψ ∈ .
m→∞
Next we consider the expectation values ϕ ⊗nl (U ∗⊗n pnl U ⊗n ) for any U ∈ Ul (, ϕ) and n ∈ N. From the assumed invariance of Dϕ ⊗l with respect to the unitary transformations given by elements of Ul (, ϕ) we conclude ϕ ⊗nl (U ∗⊗n pnl U ⊗n ) = ϕ ⊗nl (pnl ),
∀U ∈ Ul (, ϕ).
(33)
The dimension of the symmetric subspace SYM(A(l) , n) := span{A⊗n : A ∈ A(l) } is upper bounded by (n + 1)dim A , which leads to the estimate (l)
2l
tr p nl ≤ (n + 1)d · tr pnl .
(34)
Using (33), (34) and (31) we obtain 1 1 log ϕ ⊗m (pm ) ≤ log ϕ ⊗nl (pnl ) m nl 1 2l ≤ log((n + 1)d · ϕ ⊗nl (pnl )) nl (l) n (d 2l + d l ) log(n + 1) − S(, ϕ) + ηl + . ≤ nl l
(35)
For fixed l the upper bound above converges to −S(, ϕ) + ηl , for n → ∞. Choosing l sufficiently large, ηl becomes smaller than δ. This proves the upper bound. 4. Two Examples 1. Consider a quantum system where C2 is the underlying Hilbert space and let v, w be two different non-orthogonal unit vectors in C2 . Let ψ ⊗n be the product state on ⊗n , where p is the projection onto the one-dimen(B(C2 ))⊗n with density operator pw w 2 sional subspace in C spanned by w. Further let δ ≥ 0 and denote by ϕδ the state on B(C2 ) corresponding to the density operator (1 − δ)pv + δpw . It seems rather clear that for the case of the n−block state ψ ⊗n a reasonable choice of a quantum empirical distributions (states) is pw (or more general the underlying one-site state in the case of a stationary product state). So, when trying to define typical projectors via empirical states and to use these in analogy to the classical Sanov’s theorem, the n-block typical projec⊗n . tion p(n) for the set = {ψ} ⊆ S(B(C2 )) would be expected to fulfill p (n) ≥ pw ⊗n (n) ⊗n ⊗n 2 n 2n Then we have ϕδ (p ) ≥ ϕδ (pw ) = (δ +(1−δ)v, w ) ≥ v, w . On the other
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hand, the relative entropy of the density operator pw with repect to (1 − δ)pv + δpw can be made arbitrary large by choosing δ small but positive. This shows that, in contrast to the classical situation, when relying on empirical states the relative entropy rate is not an accessible separation rate (which can be at most −2 log |v, w|) . We might simplify the argument by saying that though the relative entropy of ψ with respect to ϕ is infinite the separation rate using empirical distributions remains bounded. But choosing p(n) as p(v ⊗n )⊥ , where (v ⊗n )⊥ denotes the orthogonal complement of the vector v ⊗n in (C2 )⊗n yields ϕ ⊗n (p (n) ) ≡ 0 and ψ ⊗n (p (n) ) → 1, hence the separation rate can in fact be made infinite when choosing the typical projector in another way. 2. A slightly more involved example shows that, again in contrast to the classical case, there is in general no universal choice of the separating projection, i.e. the separating projection has to depend upon the reference state ϕ. This time we will refer directly to the infinite relative entropy case and leave the simple ’smoothening’ argument which leads to a finite entropy example to the reader. Let v and w be two orthogonal unit vectors in C2 . Let be the set of pure states ϕt on B(C2 ) corresponding to the vectors vt := cos t · v + sin t · w, t ∈ [−T , T ], π2 > T > 0 and = {ψ}, where ψ is the pure state corresponding to w. Assume there is a typical projector p (n) for ψ ⊗n separating it from each ϕt⊗n super-exponentially fast. This should be valid for an universal projector since all the relative entropies S(ψ, ϕt ) are infinite. Let SYM(n) ⊂ (C2 )⊗n be the symmetrical n-fold tensor product of C2 . Without any loss of the generality we may choose p(n) ≤ pSYM(n) since all vt⊗n as well as w⊗n belong to SYM(n). Observe that the existence of p (n) (with the desired property) implies the existence of at least one (sequence of) unit vectors xn in SYM(n) such that xn , vt⊗n tends to zero super-exponentially fast uniformly in t. Choose an orthonormal basis in SYM(n) 1/2 /n! π∈PERM(n) Uπ (w ⊗k ⊗ v ⊗(n−k) ), where PERM(n) is the group by en,k := nk of n-Permutations and Uπ is the unitary operator which interchanges the order in the tensor according to π. Representing vt⊗n in that basis yields the numerical vec product n 1/2 tor nk (sin t)k (cos t)n−k . So the question is whether there exists a sequence k=0 1/2 (tan t)k tends to of unit vectors xn = (xn,k ) such that supt∈[−T ,T ] (cos t)n k xn,k nk n zero super-exponentially fast. Observe that the factor (cos t) is bounded from below by (cos T )n and can be omitted since it goes to zero only exponentially fast. Moreover, if we −1/2 xn = (xn,k nk ) we change its norm only by an at most exponentially replace xn by small factor (the maximum of the binomial coefficient is of exponential order 2n ). So we may simplify the problem by asking whether there is a sequence of unit vectors xn which has a super-exponentially decreasing inner product with the numerical vectors ((tan t)k )k=0,1,... ,n , uniformly in t ∈ [−T , T ]. This can be excluded: Let n be odd and consider the set of values tm = arctan((1 − 2 m ) · tan T ), m = 0, 1, . . . , n. Even for n this finite set of values we have necessarily supm k xn,k (tan tm )k = supm k xn,k ((1− k k k 2m n )) (tan T ) tending to zero at most exponentially fast. Ideed, the factor (tan T ) can be m k n omitted as before. Let Vn be the Vandermonde matrix (((1−2 n )) )m,k=0 . Then the L∞ norm of the vector Vn xn can be estimated by a sub-exponential factor times its L2 -norm, π π 1 and by [9], Example 6.1 the least singular value of Vn behaves like π e 4 e−n( 4 + 2 ln 2) . Acknowledgements. This work was supported by the DFG via the project “Entropie, Geometrie und Kodierung großer Quanten-Informationssysteme”, by the ESF via the project “Belearning” at the TU Berlin and the DFG-Forschergruppe “Stochastische Analysis und große Abweichungen” at the University of Bielefeld.
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References 1. Bjelakovi´c, I., Krüger, T., Siegmund-Schultze, Ra., Szkoła, A.: The Shannon-McMillan theorem for ergodic quantum lattice systems. Invent. Math. 155(1), 203–222 (2004) 2. Bjelakovi´c, I., Krüger, T., Siegmund-Schultze, Ra., Szkoła, A.: Chained typical subspaces-A quantum version of Breiman’s theorem. http://arxiv.org/list/quant-ph/0301177, 2003 3. Bjelakovi´c, I., Szkoła, A.: The data compression theorem for ergodic quantum information sources. Quant. Inform. Proc. 4(1), 49–63 (2005) 4. Bjelakovi´c, I., Deuschel, J.-D., Krüger, T., Seiler, R., Siegmund-Schultze, Ra., Szkoła, A.: A Sanov type theorem for ergodic probability measures and its quantum extension. In preparation 5. Bjelakovi´c, I., Siegmund-Schultze, Ra.: An Ergodic Theorem for the Quantum Relative Entropy. Commun. Math. Phys. 247, 697–712 (2004) 6. Bjelakovi´c, I., Siegmund-Schultze, Ra.: A New Proof of the Monotonicity of Quantum Relative Entropy for Finite Dimensional Systems. http:arxiv.org/list/quant-ph/0307170, 2003 7. Cover, T.M., Thomas, J.A.: Elements of Information Theory. New York: John Wiley and Sons, 1991 8. Deuschel, J.-D., Stroock, D.W.: Large Deviations. Boston: Acad. Press, 2001 9. Gautschi, W.: Norm estimations for inverses of Vandermonde matrices. Numerische Mathematik, 23, 337–347 (1975) 10. Hiai, F., Petz, D.: The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability. Commun. Math. Phys. 143, 99–114 (1991) 11. Jozsa, R., Schumacher, B.: A New Proof of the Quantum Noiseless Coding Theorem. J. Mod. Optics 41(12), 2343–2349 (1994) 12. Kaltchenko, A., Yang, E.H.: Universal Compression of Ergodic Quantum Sources. Quant. Inf. and Comput. 3, 359–375 (2003) 13. Ogawa, T., Nagaoka, H.: Strong Converse and Stein’s Lemma in Quantum Hypothesis Testing. IEEE Trans. Inf. Th. 46(7), 2428–2433 (2000) 14. Ohya, M., Petz, D.: Quantum entropy and its use. Berlin-Heidelberg-New York: Springer-Verlag, 1993 15. Sanov, I.N.: On the probability of large deviations of random variables. Mat. Sbornik 42, 11–44 (1957) 16. Shields, P.C.: Two divergence-rate counterexamples. J. Theor. Prob. 6, 521–545 (1993) 17. Takesaki, M.: Theory of operator algebras I. Berlin-Heidelberg-New York: Springer-Verlag, 1979 Communicated by M.B. Ruskai
Commun. Math. Phys. 260, 673–709 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1431-5
Communications in
Mathematical Physics
CMC-Slicings of Kottler-Schwarzschild-de Sitter Cosmologies Robert Beig1 , J. Mark Heinzle2 1
Institute for Theoretical Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria. E-mail: [email protected] 2 Max-Planck-Institute for Gravitational Physics, Am M¨ uhlenberg 1, 14476 Golm, Germany. E-mail: [email protected] Received: 3 January 2005 / Accepted: 23 February 2005 Published online: 20 September 2005 – © Springer-Verlag 2005
Abstract: There is constructed, for each member of a one-parameter family of cosmological models, which is obtained from the Kottler-Schwarzschild-de Sitter spacetime by identification under discrete isometries, a slicing by spherically symmetric Cauchy hypersurfaces of constant mean curvature. These slicings are unique up to the action of the static Killing vector. Analytical and numerical results are found as to when different leaves of these slicings do not intersect, i.e. when the slicings form foliations.
1. Introduction Consider a globally hyperbolic spacetime which is spatially compact. One can ask the question whether there exists a slicing by Cauchy surfaces of constant mean curvature (in short: CMC-slicing) and what are its properties. Since CMC-slicings have a wide variety of uses in general relativity, this question has received a lot of attention, we refer to [9] and [1] for recent overviews. In the present paper we study this problem for a class of spatially compact spacetimes, which are quotient spaces of the Kottler-Schwarzschild-de Sitter family of spacetimes [7] under certain discrete subgroups of their isometry group. These spacetimes can be viewed as models of cosmological spacetimes containing a black hole. Studies of CMCslicings which we are aware of treat cases where the spacetime has — in the future and/or past — either an all-encompassing crushing singularity or is geodesically complete, whereas in Kottler-Schwarzschild-de Sitter both types of asymptotic behavior coexist. Furthermore, the Kottler-Schwarzschild-de Sitter spacetimes, due to the presence of a positive cosmological constant, violate the so-called timelike convergence condition (strong energy condition) which most papers assume. Before we state the main results of the present work, let us define and describe the family of Kottler-Schwarzschild-de Sitter cosmologies we consider:
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The Kottler-Schwarzschild-de Sitter metric reads ds 2 = −V dt 2 + V −1 dr 2 + r 2 d2 ,
where V = V (r) = 1 −
2M r 2 − . (1) r 3
The cosmological constant > 0 and the constant M > 0 are required to satisfy 9M 2 < 1 ;
(2)
then the function V (r) is positive in the interval (rb , rc ), and V (r) = 0 at the “black hole horizon” r = rb and at the “cosmological horizon” r = rc . The region rb < r < rc is a static region of the spacetime with Killing vector ξ = ∂t . (When 9M 2 > 1, there are no horizons, the spacetime is not static but homogeneous.) It is straightforward to see that 1 3 2M < rb < 3M < √ < rc < √ .
(3)
In the limit where goes to zero, the spacetime metric tends to Schwarzschild and rb tends to 2M. √ In the limit where M goes to zero, the metric becomes de Sitter and rc tends to 3/ . The static region of the spacetime has an analytic extension reminiscent of the Kruskal extension of the Schwarzschild spacetime and the de Sitter spacetime. A common way of depicting the arising spacetime uses two charts which cover the regions 0 < r < rc and rb < r < ∞. The conformal compactification of the region 0 < r < rc is depicted in Fig. 1(a); it contains the curvature singularity r = 0; the conformal compactification of rb < r < ∞ is shown in Fig. 1(b). The overlap of the charts is rb < r < rc . As is well-known, the constructed spacetime corresponding to the union of Figs. 1(a) and 1(b) can be smoothly (in fact analytically) extended in a “periodic–in–r”-fashion. Thereby one obtains an inextendible, globally hyperbolic spacetime of topology R×R×S 2 satisfying Gµν + gµν = 0, see Fig. 2. We call this spacetime the Kottler-Schwarzschild-de Sitter spacetime (KSSdS). On KSSdS there exists an isometric action of R×SO(3). The dashed lines in Figs. 1(a) and 1(b) are orbits under the R-factor in this action, i.e. under the static Killing vector ξ = ∂t . Henceforth this action will be called “Killing flow” for brevity. Note that the Killing vector ξ is globally defined; it is null on the Killing horizons r = rb and r = rc which emanate from the bifurcation 2-spheres at which ξ vanishes. Furthermore there exist discrete isometries (“reflections”) leaving fixed the hypersurfaces t = T = const, which are given via T + t → T − t. While r is globally defined on KSSdS, t blows up on the Killing horizons. In the static region or in the black hole, white hole, or cosmological regions, (r, t) forms a coordinate system. The solid lines in Figs. 1(a) and 1(b) represent hypersurfaces t = const, which are totally geodesic as fixed point sets of the discrete isometries. There exist two kinds: spacelike and timelike t = const hypersurfaces; the latter we call t = const cylinders. By virtue of the periodicity in r, the t = 0 cylinders in two adjoining copies of the region 0 < r < rc can be identified, which results in a smooth spacetime of topology R × S 1 × S 2 , which we call the cosmological Kottler-Schwarzschild-de Sitter spacetime KSSdS[0]. The spacetime KSSdS is the universal covering of KSSdS[0]. More generally, let T ∈ R and identify points of equal radius r on a t = 0 cylinder and a tilted t = 2T cylinder in an adjacent copy of the region 0 < r < rc on the r.h. side, see Fig. 2. Thereby we obtain a whole family of inextendible, globally hyperbolic, cosmological spacetimes, which we call KSSdS[T].
CMC-Slicings of KSSdS Cosmologies
675
r
r
− ∞
r=0 ∞ = rb
,t
t> 0
t =0
= ,t
r
∞
∞
=
=
−
rc
,t
=
=
rc
,t
0 t<
rb
t >0 t =0
t >0
t <0
r
∞
r
=
=
t <0 t =0
=
=
rb
,t
,t
,t
rb
=
r
∞
=
−
∞
=
=
r
,t
rc
rc
∞
=
−
r
=
r=0
(a) 0 < r < rc r=∞ ∞ −
t< 0
= ,t
= ,t
t =0
rc
=
=
rb
,t
,t
=
−
∞
∞
=
rb
rc
0 t>
=
t <0
r
r
=
∞
r
r
t >0 t =0
t =0
t <0
t >0
r ∞ = ,t rb r
r
∞
=
=
−
∞
rc
=
=
,t
,t
,t
=
rc
rb
−
=
∞
r
=
r=∞
(b) rb < r < ∞ Fig. 1. The figures show the compactified regions 0 < r < rc and rb < r < ∞. Solid lines represent hypersurfaces t = const, dashed lines are hypersurfaces r = const r=∞
rb r
rc =
t = 2T
rb = r
r
= r
rc =
rb
=
rc
rb = r
r
rb
t =0
=
=
r
rc
rb
rc
=
=
=
r
r
r r
rc
rc
rb
rb
=
=
=
r
r
=
=
r
r=0
r
r
rc
r=0
identification
Fig. 2. The compactified Kottler-Schwarzschild-de Sitter spacetime KSSdS[T]
Note that KSSdS[T] is a smooth, in fact analytic, spacetime: consider a neighborhood of the t = 0 cylinder in KSSdS. Via the Killing flow this neighborhood is isometric to a neighborhood of the t = 2T cylinder, hence the hypersurfaces t = 0 and t = 2T agree not only in their induced first and second fundamental forms, but also in all higher derivatives of the fundamental forms. The identification of the two hypersurfaces thus results in a smooth manifold. Note that, while the spacetime KSSdS[0] is time-symmetric,
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KSSdS[T] with T = 0 is not; however, KSSdS[T] and KSSdS[−T] differ by time orientation only. The spacetimes KSSdS[T] are not the only cosmological spacetimes which arise as quotient spaces from KSSdS. Firstly, we note that it is not necessary to identify t = 0 with a t = const cylinder in an adjoining copy of the region 0 < r < rc ; indeed, an arbitrary number of intervening copies between the identified copies is possible. However, the arising spacetimes do not exhibit different structures as far as the properties of CMC-slicings are concerned. Secondly, we may identify, in the black/white hole or in the future/past cosmological region, points mapped to each other by a discrete subgroup of the action under ξ . Thereby we obtain cosmological spacetimes that are completely different from the class KSSdS[T] considered here. These spacetimes will be treated in a forthcoming work by one of us (J.M.H.) [5]. In this paper, a slicing denotes a smooth family of smooth (spacelike) hypersurfaces. A parametrization of a slicing is a smooth map : I × → spacetime, where is a 3-manifold and I ⊆ R, such that for all τ ∈ I , (τ, ·) is an embedding. We require that (τ, ) is a hypersurface of the slicing for all τ ∈ I , i.e. by the parametrization of the slicing the hypersurfaces are represented as level sets τ = const. Note that a slicing is a foliation iff the map is a diffeomorphism onto its image. The content of the paper is as follows. In Sect. 2 we study spherically symmetric compact CMC-slices in terms of the associated initial data sets. We find that these compact CMC-initial data sets are parametrized by two constants (K, C), which lie in a bounded connected open subset KC0 of R2 . The constant K is the mean curvature of the slice, i.e. the trace of the second fundamental form. The interpretation of the constant C is less immediate (see the note following (6)). Suffice it to say that initial data generated by (K, C) is umbilical, i.e. the extrinsic curvature is proportional to the three-metric, iff C = 0. The parameter space KC0 is investigated in some detail in Appendix A. In Sect. 3 we discuss the embedding of the compact initial data sets into the cosmological spacetimes KSSdS[T]: we prove that each compact CMC-initial data set is embeddable as a Cauchy CMC-hypersurface into KSSdS[T] for a particular value of T and that the embedding is unique modulo the Killing flow. Sections 4 and 5 are concerned with the formulation and the proof of the main results of the present paper. One main theorem can be stated in an informal manner as follows: Each spacetime KSSdS[T] contains a unique non-trivial slicing of Cauchy CMC-hypersurfaces. (Here, a CMC-slicing is called trivial, if both K and C are constant along the slicing, and uniqueness is again understood modulo the Killing flow.) The first step in proving this theorem is undertaken in Sect. 4, where we show, using the implicit function theorem, that if the spacetime KSSdS[T] contains a compact CMC-hypersurface, then it evolves into a unique CMC-slicing. Crucial for this is the analysis of a (linear) ordinary differential equation, whose solution is interpreted as the lapse function of the slicing. Due to the violation of the timelike convergence condition, this analysis is quite involved and thus deferred to Appendix B. We find that the CMC-slicing in KSSdS[T] can be represented by a curve in the parameter space KC0 , which in turn is given as the T-level set of a function T (K, C) on KC0 . Thus there exists an interplay between a global but finite dimensional picture, where CMC-slicings are viewed in terms of their representation in KC0 on the one hand, and a local but essentially infinite dimensional view of slicings in spacetime on the other hand. Based on these ideas, in the second step in the proof of the theorem, in Sect. 5, we show that each spacetime KSSdS[T], T arbitrary, contains a compact CMC-slicing and that this slicing is unique.
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In another theorem in Sect. 5 we √ describe the asymptotic behavior of the CMC-slicings: Along each slicing K tends to ± 3 in the future, resp. in the past, C tends to explicitly known values that depend on and M. In spacetime the hypersurfaces of the slicing √ √ approach r = (1/ ) 1 − 1 − 3 M in the black hole (resp. white hole), and r = ∞ in the future (resp. past) cosmological region (see Fig. 8). An essential ingredient in the proof of the theorem is an asymptotic analysis of the function T . Finally, in Sect. 5 and in Appendix C, we discuss whether the compact CMC-slicings in the spacetimes KSSdS[T] are foliations. We prove that each slicing is a foliation at least during some time of its evolution. Moreover, if |T| is sufficiently large, then the slicing cannot be a foliation for all times. In addition, we provide solid numerical evidence that there are values for (, M) such that, if |T| is small, the CMC-slicing is a foliation everywhere. In this work we can, and of course do, make use of the spherical symmetry of the spacetimes. The quantities we are seeking are all “essentially explicit” in this sense: they are either given by quadratures of algebraic functions, or solutions to algebraic equations, or level sets of functions which are in turn given by quadratures, and combinations of the above. Still only a small part of our task can be performed by a direct investigation of these expressions. Rather we require a somewhat delicate interplay between the analysis of the explicit quantities and the geometric analysis. 2. CMC-Data In this section we investigate spherically symmetric CMC-initial data sets. We find that there exists a two-parameter family of compact CMC-data; we analyze the parameter space, and we discuss the main geometric properties of the CMC-data sets in dependence on the parameters. Consider a three-dimensional Riemannian manifold ∼ = J × S 2 endowed with a spherically symmetric 3-metric gij dx i dx j = dl 2 + r 2 (l)(dϑ 2 + sin2 ϑ dϕ 2 ) ;
3
(4)
the coordinates ϑ, ϕ are usual angular coordinates, l is a “radial” coordinate which takes values in J , which is (an open interval of) R, or J ∼ = S 1 . Let there be given a symmetric tensor Kij , the second fundamental form, such that the mean curvature K = 3gij K ij = const .
(5)
By (locally) solving, along the lines of [3], the vacuum constraints with positive cosmological constant we obtain that (, 3gij , Kij ) is a spherically symmetric CMC-initial data set, iff there exists a constant C such that K K 2C C 2 dl r(l)2 (dϑ 2 + sin2 ϑ dϕ 2 ) . (6) + + − Kij dx i dx j = 3 r(l)3 3 r(l)3 Note that Kij is of the form Kij = (K/3) 3gij + C LTT ij , i.e. it is the sum of a constant trace plus C times a tensor that coincides with the unique spherically symmetric transverse traceless tensor w.r.t. 3g. The initial data set is umbilical, i.e. Kij ∼ gij , iff C = 0. The function r(l) is required to satisfy r 2 Kr C 2 2M r 2 = 1 − − + − 2 =: D(r) , (7) r 3 3 r
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where the prime denotes the derivative w.r.t. l. The constant M in (7) coincides with the mass M appearing in (1), which is a consequence of the subsequent considerations. An initial data set (, 3gij , Kij ) is said to admit the Killing initial data (or KID) (α, Xi ), if (α, Xi ) lies in the kernel of the (overdetermined) operator given by the adjoint of the linearized constraint operator (see [4], also [2]). The KID-condition, in the presence of , is equivalent to 2αKij + 2D(i Xj ) = 0 ;
(8)
Di Dj α + LX Kij − α(3Rij + KKij − 2Kil Kj l − 3gij ) = 0,
(9)
together with
which, provided that α = 0, is in turn equivalent to the statement that the “Killing development”, i.e. the stationary metric gµν dx µ dx ν = (−α 2 + X 2 )dτ 2 + 2Xi dτ dx i + 3gij dx i dx j
(10)
satisfies Gµν + gµν = 0. The present CMC-initial data set does in fact admit a KID, namely (αξ , Xξi ) given by
αξ = r (l)
Xξi ∂i
,
∂ C Kr(l) ∂ − . = xξ (l) = − 2 ∂l 3 r(l) ∂l
(11)
The development of the initial data set thus results in a spacetime with Killing vector ξ µ , r 2 2M µ ξ µ = αξ n µ + X ξ ∂ µ , − . (12) ξ µ ξµ = −αξ2 + Xξ2 = − 1 − r 3 ξ
Here, nµ denotes the unit normal of in the spacetime, Xξ2 = Xξi Xi . The KID is “static”: from ξ Xj ] D[i =0 (13) −αξ2 + Xξ2 it follows that ξ µ is hypersurface orthogonal. In the Killing development of the initial data, is given by τ = 0, and the metric reads ξ
gµν dx µ dx ν = (−αξ2 + Xξ2 )dτ 2 + 2Xi dτ dx i + 3gij dx i dx j .
(14)
By using r and introducing the coordinate t through ξ
dτ = dt −
Xi
−αξ2 + Xξ2
dx i = dt − V −1
Kr(l) C − 3 r(l)2
dl,
(15)
we recover the original Schwarzschild-de Sitter metric (1), −1 r 2 r 2 2M 2M gµν dx µ dx ν = − 1 − − dt 2 + 1 − − dr 2 + r 2 d2 . r 3 r 3 (16)
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We record for later use the identity ∇µ t = −V −1 ξµ ,
(17)
which follows from the previous discussion. As another particular consequence we conclude that the constant M introduced in (7) is to be interpreted as the mass M appearing in the Kottler-Schwarzschild-de Sitter metric. In the following we consider the cosmological constant and the mass M as given constants. A CMC-initial data set (, 3gij , Kij ) arises from the choice of a pair (K, C), by which D(r) and thus r(l) is defined (modulo an irrelevant translational freedom in l), cf. (7). Definition 2.1. We define the parameter space of compact CMC-initial data by KC := {(K, C) | (K, C) generates compact CMC-data (, 3gij , Kij ) ◦ with ∼ = S1 × S2} .
(18)
Proposition 2.2. The parameter space KC is the disjoint union of three open connected domains, KC = KC0 ∪ KC1 ∪ KC−1 .
(19)
KC0 is the connection component of (K, C) = (0, 0), it is invariant under inversion. For proofs we refer to App. A. In this paper, we focus on compact CMC-initial data generated by (K, C) ∈ KC0 . The space KC0 is depicted in Fig. √ 4; it is enclosed by the curves Ct and Cb , and the vertical straight lines K = ± 3. (In the following we refrain from making a distinction between the curves Cb,t and the functions Ct (K), Cb (K) = −Ct (−K), that parametrize the curves.) Some properties of the functions Cb,t (K) are discussed in App. A. If and only if (K, C) ∈ KC, the function D(r) exhibits the form depicted in Fig. 3(a): in particular, D(r) possesses two positive (simple) zeros, rmin and rmax , such that D(r) > 0 in the interval (rmin , rmax ). Accordingly, when viewed over r ∈ [rmin , rmax ],
D(r)
r
r+
r r rone
rmin
rmax rmin
r−
(a)
rmax
(b)
Fig. 3. For (K, C) ∈ KC, D(r) is positive between two positive zeros, rmin /rmax , and dD/dr does not vanish at rmin /rmax . Then D(r) gives rise to a (smooth) closed curve in the (r, r )-plane, and r(l) becomes a periodic function
680
R. Beig, J. M. Heinzle C K=
) → ← Ct (K
√ 3
√
√ K = − 3
√
K
) → ← C b (K
√ Fig. 4. KC0 for = 1, M = 1/4; it is enclosed by Cb,t (K) and K = ± 3
√ r = ± D(r) describes a closed curve, cf. Fig. 3(b), so that r(l) becomes a periodic function that oscillates between rmin and rmax ; we denote the period by 2L,
rmax L= D −1/2 (r) dr .
(20)
rmin
Without loss of generality we assume that r(0) = rmin
so that
r(±L) = rmax ;
(21)
it follows that r(l) is an even periodic function, which is implicitly given through
r l(r) = ± D −1/2 (ˆr ) d rˆ . (22) rmin
By the natural identification of l = −L and l = L, the domain of the function r(l) becomes S 1 , so that the CMC-initial data (, 3gij , Kij ) is compact with ∼ = S1 × S2. Remark 2.3. From Fig. 3(a) we see that a pair (K, C) that generates compact CMC-data, in general also gives rise to non-compact CMC-data, where r(l) ranges in (0, rone ]. A CMC-data set of this type is embeddable in KSSdS as a hypersurface that runs into the singularity. A detailed classification of all possible types of CMC-data sets and their embeddings in KSSdS[T] and other cosmological KSSdS-spacetimes will be presented in [5]. When (K, C) ∈ ∂(KC), the profile of the function D(r) is a borderline case of the profile 3(a). On the boundaries Cb,t , the function has the form 5(a), i.e. the zero rmin √ is a double zero. The solution of r = ± D(r) that is relevant for our purposes is r(l) ≡ const = rmin : by identifying l = l0 with l = l1 for any l0 , l1 , we obtain a√ compact CMC-initial data set (, 3gij , Kij ), ∼ = S 1 × S 2 . On the boundaries K = ± 3, D(r) possesses a simple zero rmin , but “rmax = ∞”, i.e. D → 1 (r → ∞), see Fig. 5(b). In this case, no compact CMC-data is generated. The features of D(r) described above occur simultaneously on the intersections of the boundaries. For every compact CMC-initial data set (S 1 ×S 2 , 3gij , Kij ) associated with (K, C) ∈ KC0 , the radius r of the spheres of symmetry varies between values rmin , rmax . We conclude the section by discussing rmin (K, C) and rmax (K, C) in dependence on (K, C) ∈ KC0 .
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D(r)
D(r)
r
r
√ (b) K = ± 3
(a) (K, C) ∈ Cb,t
Fig. 5. The profiles of D(r) on ∂(KC0 )
Most importantly, we note that rmin ≤ rb and that rmin = rb iff (K, C) is such that C = (rb3 /3)K. This “line of maximal rmin ” divides KC0 in two regions, an upper 2 ] < 0, and a lower (right) half with (left) half, characterized by [Krmin /3 − C/rmin 2 [Krmin /3 − C/rmin ] > 0, see Fig. 10(a). The function rmin assumes its global mini√ √ mum on KC0 at the point (K, C) = ( 3, Cb ( 3)) and at the reflected point; the minimal value is given by (23). rmin decreases/increases along the boundaries of KC0 as depicted in Fig. 10(a). rmax ≥ rc and rmax = rc iff (K, C) is such that C = (rc3 /3)K. This “line of minimal rmax ” divides KC0 in two regions, an (upper) left half, where 2 ] < 0 and a (lower) right half, where [Kr 2 [Krmax /3 − C/rmax √ /3 − C/rmax ]√> 0, √ max ¯ as K → ± 3; see Fig. 10(b). rmax is unbounded on KC0 , it diverges like 3/ ¯ = − K 2 /3. Both rmin and rmax are constant along straight lines in KC0 , see Fig. 11. rmin can be given explicitly, √ √ √ 1 rmin = √ for K = 3, C = Cb,t ( 3). (23) ±1 ∓ 1 ∓ 3 M , Here, the upper sign applies to Ct , the lower to Cb . 3. Embeddings In this section we investigate embeddings of compact CMC-initial data as CMC-hypersurfaces in Schwarzschild-de Sitter spacetime. Let (K, C) ∈ KC0 . The pair generates a compact CMC-initial data set ˜ ∼ ( ∼ = R × S 2 , 3gij , Kij ) = S 1 × S 2 , 3gij , Kij ). Consider the universal covering ( of the data by regarding r(l) as a periodic even function on R. The initial data set ˜ 3gij , Kij ) is embedded as a CMC-hypersurface S˜ in KSSdS via (,
l r = r(l)
t = t (l) :=
V 0
−1
ˆ (r(l))
ˆ C Kr(l) − ˆ2 3 r(l)
d lˆ ,
(24)
which follows from (15). The integral is understood in the principal value sense, so that t (l) is well-defined for all l with V (r(l)) = 0; this suffices to uniquely define the embedded hypersurface in KSSdS. (Alternatively, the embedding can be given in Kruskal type coordinates.) If (K, C) is such that rmin = rb , then V −1 is singular at l = 0, but also (Kr/3 − C/r 2 ) = 0 when l → 0, cf. Fig. 10(a). Using de l’Hospital’s rule
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R. Beig, J. M. Heinzle
we see liml→0 V −1 (r(l))[Kr(l)/3 − C/r 2 (l)] = −V −1 (rb )(K/3), i.e. the integrand is bounded as l → 0. Similarly, the integrand is always bounded as l → L. ˜ inteThe CMC-hypersurface S˜ defined by (24) is not the unique embedding of : grating (15) leaves the freedom of choosing an integration constant so that t is replaced by t − const in (24). The one-parameter freedom is associated with the Killing flow, hence, modulo the Killing isometries the embedding is indeed unique. Let (K, C) ∈ KC0 . Then r(l) oscillates between rmin ≤ rb and rmax ≥ rc , therefore S˜ is a Cauchy hypersurface in KSSdS, see Fig. 6. The (future pointing) unit normal of S˜ is given by ∂ ∂ C Kr nµ ∂µ = r V −1 + − 2 ; (25) ∂t 3 r ∂r it is straightforward to check ∇µ nµ = K. The CMC-hypersurface S˜ passes through 2 ) > 0 and through the black hole region if the white hole region if (Krmin /3 − C/rmin 2 2 ) = 0, r (Krmin /3−C/rmin ) < 0; when (Krmin /3−C/rmin min = rb and the hypersurface passes through the bifurcation sphere; compare with Fig. 10(a). Similarly, the hypersur2 ) > 0 face passes through the future [past] cosmological region if (Krmax /3 − C/rmax 2 [Krmax /3 − C/rmax < 0], cf. Fig. 10(b). These relations guarantee that S˜ cannot oscillate between the black hole and the white hole (or the future and the past cosmological regions): either S˜ passes through the black hole regions or through the white hole regions for all l = 2nL, n ∈ N, where r(l) = rmin . However, it will occur that S˜ runs through the black hole region and through the past cosmological horizon (or vice versa). This can be seen from the combination of Figs. 10(a) and 10(b). The embedding of a compact CMC-initial data set (, 3gij , Kij ), ∼ = S 1 × S 2 , as a CMC-hypersurface S in a cosmological spacetime KSSdS[T] is more delicate. Consider (24) and make the following Definition 3.1. Define T := t (L), i.e.
L T =
V
−1
(r(l))
Kr(l) C − 3 r(l)2
rmax Kr C −1 −1/2 dl = V D (r) − 2 dr. (26) 3 r rmin
0
r=∞
r=0
r=0
n
r=
n
r mi
r=
r mi
t = 2T
t =0
identification
Fig. 6. The embedding of the compact CMC-initial data with K = 1 and C = 0.1 in KSSdS[T] (with = 1, M = 1/4). The hypersurface is not null at the horizon rc : for the figure we have used the coordinate system of the region 0 < r < rc and extended it to 0 < r < ∞ to obtain global coordinates; however, these coordinates break down at r = rc
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683
Like rmin and rmax , the function T depends on the pair (K, C) ∈ KC0 characterizing the CMC-data. For (K, C) ∈ Cb,t , T is not canonically defined; each choice of l0 , l1 , which leads to compact data, cf. Sect. 2, is associated with a different value of T . Proposition 3.2. Let (K, C) ∈ KC0 . Then the associated compact CMC-initial data ( ∼ = S 1 × S 2 , 3gij , Kij ) is embeddable as a CMC-hypersurface S in the spacetime KSSdS[T] if and only if T = T . The hypersurface is a Cauchy hypersurface, and the embedding is unique modulo the Killing flow. ˜ of the data and the embedded hypersurface S˜ Proof. Consider the universal covering as given by (24). The hypersurface is invariant under the two discrete isometries t → −t and 2T + t → 2T − t of the spacetime; this is because r(l) is even about l = 0 and even about l = 2L according to its periodicity. Since the unit normal vectors of S˜ at l = 0 and l = 2L must be invariant under the respective isometries as well, they must be tangential to the fixed point sets t = 0 and t = 2T of the isometries: we conclude that nµ ∂µ ∝ ∂ at l = 0 and l = 2L. The identification of t = 0 with t = 2T in KSSdS thus results in a CMC-hypersurface S in KSSdS[T ] which is isometric to by construction. The proof of the remaining claims is trivial. The argument used in the proof will re-appear in the proof of Theorem 4.1; alternatively, we could have used (25). Remark 3.3. Consider (the universal covering of) the CMC-initial data generated by (K, C) ∈ Cb,t ⊂ ∂(KC). Since the data is characterized by a function r(l) ≡ const, the embedding is a r = const hypersurface, which is contained in the black/white hole region of KSSdS[T] (for arbitrary T). The compact interpretation of the data set, which is obtained by an identification of l = l0 with l = l1 for any l0 , l1 , is not naturally embeddable in KSSdS[T]. 4. Slicings Proposition 3.2 states that there exist spacetimes KSSdS[T] containing a compact CMChypersurface. The aim of this section is to investigate whether these spacetimes contain CMC-slicings, i.e. smooth families of CMC-hypersurfaces. Any CMC-hypersurface can be evolved into a CMC-slicing by the Killing flow; however, this CMC-slicing is trivial in the sense that K and C are constant along the slicing. The existence of non-trivial slicings is shown in the following theorem: any compact CMC-hypersurface of KSSdS[T] evolves into a compact CMC-slicing, along which the mean curvature K is monotonic; furthermore, this slicing is necessarily unique modulo the Killing flow. Theorem 4.1. Consider a spacetime KSSdS[T] that contains a compact CMChypersurface S. Then there exists a unique local slicing of KSSdS[T] by hypersurfaces Sτ , τ ∈ (−τ¯ , τ¯ ), such that i. S0 = S, ii. Sτ is a compact CMC-hypersurface for all τ ∈ (−τ¯ , τ¯ ), iii. Sτ is reflection symmetric for all τ . Along the slicing Sτ the mean curvature is a strictly monotonic function, i.e. iv. K(τ ) is strictly monotonic.
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R. Beig, J. M. Heinzle
Furthermore, every slicing Sτ that satisfies (i) and (ii) arises from Sτ by combining the flow of Sτ with an appropriate admixture of the Killing flow. Remark 4.2. The requirement (iii) is a convenient way of fixing a representative within the equivalence class of slicings that satisfy (i) and (ii). The compact CMC-hypersurface S contains a totally geodesic 2-sphere at r = rmin , which evolves into a cylinder represented by τ → rmin (τ ). The slicing Sτ is reflection symmetric if this cylinder is totally geodesic, i.e. coincides with t = const; without loss of generality we always assume reflection symmetry w.r.t. t = 0. Proof. The CMC-hypersurface S is characterized by (KS , CS ) ∈ KC0 ; the induced metric is dl 2 + rS2 (l) d2 . S can be represented by (24), it is invariant under the reflection t → −t. We introduce Gaussian coordinates in a neighborhood of S: the metric reads gµν dx µ dx ν = −dσ 2 + s 2 (σ, λ)dλ2 + r 2 (σ, λ)d2 ;
(27)
σ ∈ (−σ¯ , σ¯ ). The hypersurface S is represented by σ = 0; λ = l on S. We obtain by using the identity ∂σ gij = 2Kij and (6), s(0, λ) = 1, (∂σ s)(0, λ) =
2CS KS + 3 , 3 rS (λ)
r(0, λ) = rS (λ), CS KS rS (λ) − 2 (∂σ r)(0, λ) = . 3 rS (λ)
(28a) (28b)
We choose σ¯ so that inf |σ |<σ¯ s(σ, λ) ≥ const and inf |σ |<σ¯ r(σ, λ) ≥ const ∀λ. The normal vector of S at λ = 0 is tangential to the cylinder t = 0, since S is invariant under the discrete isometry t → −t. The cylinder t = 0, being the fixed point set of a discrete isometry, is totally geodesic, therefore the normal geodesics passing through λ = 0 lie on t = 0, hence the hypersurfaces λ = 0 and t = 0 coincide; analogously, λ = LS corresponds to t = TS (= T). From this fact that the Gaussian coordinates are adapted to the discrete symmetry, it thus follows that s(σ, λ) and r(σ, λ) are even functions in λ for all σ ; analogously, the functions are even about λ = LS . In a neighborhood of S in KSSdS[T], a compact hypersurface can be described by the equations σ = ϕ(l) ,
λ=l,
where ϕ : S 1 → (−σ¯ , σ¯ ) .
We define the mean curvature operator K, ϕ 1 2s∂σ r ∂σ s + − 2 K[ϕ] := r (s − ϕ 2 )3/2 s2 − ϕ 2 ϕ ϕ 2r 1 + + ; rs s 2 − ϕ 2 s s2 − ϕ 2
(29)
(30)
for ϕ, the prime denotes a derivative w.r.t. l, for s and r, a derivative w.r.t. the second argument; the suppressed arguments of the functions s, r, and its derivatives, are (ϕ(l), l). For a given ϕ, K[ϕ] is a function S 1 → R describing the mean curvature of the hypersurface σ − ϕ(λ) = 0. The hypersurface S is represented by ϕ = 0; from (28) we obtain K[0] = KS . In order to show the claims of the theorem we solve the prescribed mean curvature equation K[ϕ] = K (≡ const) in a neighborhood of S.
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685
Consider the Sobolev space H 2 (S 1 ) = W 2,2 (S 1 ). The Sobolev inequalities ensure that ψ ∈ H 2 (S 1 ) implies ψ ∈ C 1,1/2 (S 1 ), where C 1,1/2 denotes the relevant H¨older space; we have the estimate ψC 1,1/2 ≤ const ψH 2 for all ψ. Since ψL∞ + ψ L∞ ≤ ψC 1,1/2 , the set H 2 (S 1 ) := {ψ ∈ H 2 (S 1 ) | ψL∞ + ψ L∞ < δ = const}
δ
(31)
k (S 1 ) denote the space of even funcis a neighborhood of the origin in H 2 (S 1 ). Let Heven k 1 k tions ψ ∈ H (S ); as a closed subspace of H (S 1 ) it is a Banach space. Finally define δH 2 (S 1 ) := H 2 (S 1 )∩ δH 2 (S 1 ). We choose δ sufficiently small, so that ψ ∞ < σ ¯ L even even and s 2 − ψ 2 L∞ ≥ const > 0. The quasilinear operator K, viewed as a map 2 0 K : δHeven (S 1 ) → Heven (S 1 )
(32)
is C 1 -differentiable. The argument is standard and can be inferred, e.g. from Thm. 4.1 in [11]. The (Fr´echet) derivative of K at a point ϕ is a linear operator, 2 0 K [ϕ] : Heven (S 1 ) → Heven (S 1 ) ,
(33)
which, for ϕ regular enough, is given by
K [ϕ](ϕ) ˙ =
(3gϕ ) + − Kϕ ij Kϕij
s s2 − ϕ 2
ϕ˙
(34)
2 (S 1 ). The expressions 3g and K are the metric and extrinsic curvature with ϕ˙ ∈ Heven ϕ ϕ of the surface σ −ϕ(λ) = 0; the Laplacian is the one associated with 3gϕ . It is convenient to obtain Eq. (34) through geometric arguments, cf. the remark following Corollary 4.5. Since ϕ ≡ 0 represents the CMC-hypersurface S we obtain
K [0] = + a
where
a(l) = −
KS2 6C 2 − 6 S . 3 r (l)
(35)
(Note that there do not exist pairs (KS , CS ) ∈ KC0 such that a(l) ≤ 0 ∀l.) The operator K [0] is elliptic, the Fredholm alternative holds. In Lemma 4.4 we show that ker K [0] is trivial, i.e. the homogeneous equation α + aα = 0 admits only the trivial solution α = 0. It follows that K [0] is an isomorphism, and we are able to apply the inverse func0 (S 1 ) and tion theorem, see, e.g. [8]: there exists an open neighborhood V of K[0] in Heven −1 δ 2 1 a unique continuously differentiable mapping K : V → Heven (S ) with the property that K[K−1 (κ)] = κ for all κ ∈ V . Let K be a smooth real function of τ ∈ R which is strictly monotonic and satisfies K(0) = KS ; let (−τ¯ , τ¯ ) be an interval such that K(τ ) ∈ V for all τ ∈ (−τ¯ , τ¯ ); note that K(τ ) is interpreted as a constant function for each τ . Then ϕτ := K−1 (K(τ ))
(36)
2 (S 1 )-functions {ϕ | τ ∈ (−τ¯ , τ¯ )}, where ϕ = ϕ uniquely defines a family of δHeven τ τ1 τ2 for τ1 = τ2 . For each τ , ϕτ is smooth, by elliptic regularity. Moreover, the mapping τ → ϕτ is continuously differentiable by construction. Therefore {ϕτ } defines a unique local slicing of KSSdS[T] by compact CMC-hypersurfaces,
Sτ := {(σ, λ, ) ∈ KSSdS[T] | σ = ϕτ (l), λ = l} ,
(37)
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R. Beig, J. M. Heinzle
τ ∈ (−τ¯ , τ¯ ); Sτ1 = Sτ2 for τ1 = τ2 ; since ϕ0 ≡ 0, S0 = S. By construction, since ϕτ is even for each τ , and since λ = 0 coincides with t = 0, Sτ is invariant under the discrete isometry t → −t for all τ . Hence, the properties (i)–(iii) are proved; also (iv) is a direct consequence of the construction. The remaining claim follows immediately from the considerations of Sect. 2 and Prop. 3.2. Remark 4.3. In the proof, K(τ ) is strictly monotonic, which does not imply that K˙ = 0 a priori. However, by an appropriate redefinition of the parameter τ , K˙ = 0 can always be achieved. Henceforth we will always adopt the convention that K˙ > 0. Lemma 4.4. Consider the equation α + aα = 0 ,
(38)
2 (S 1 ) be a solution. Then α ≡ 0. where a(l) = − K 2 /3 − 6C 2 /r 6 (l), and let α ∈ Heven
For the proof we refer to Appendix B. Corollary 4.5. The equation α + aα = K˙ possesses a unique even solution α :
S1
(K˙ = const > 0)
(39)
→ R for all (K, C) ∈ KC0 .
Proof. The equation is elliptic, the homogeneous equation possesses only the trivial solution. The Fredholm alternative holds, hence existence and uniqueness of an even periodic solution α follows. Equation (39) is obtained by differentiating the mean curvature 3g ij Kij and using the evolution equations along the slicing. The solution of (39) is the lapse function of the slicing Sτ described in Theorem 4.1: in fact, when we set
: (τ, l, ) → (σ, λ, ) = (τ, l, ) = ϕ(τ, l), l, , (40) where ϕ(τ, ·) = ϕτ (·), and note that the future unit (co)normal of the slicing Sτ is given by
s −dσ + ϕ dλ , (41) nµ dx µ = s2 − ϕ 2 we obtain α=
s s2 − ϕ 2
ϕ˙
(42)
˙ µ = αnµ + X i (where the dot denotes ∂/∂τ ). Hence (34) from the decomposition ,i and (39) coincide. The shift vector of the slicing Sτ depends on the “spatial gauge” we are imposing. Let us briefly elaborate on this issue. Consider the parametrization of the slicing
(43) : (τ, l, ) → (σ, λ, ) = (τ, l, ) = ϕ(τ, βτ (l)), βτ (l), , µ
where βτ (·) : [−LS , LS ]∼ ∼ = S 1 → S 1 is one-to-one for each τ ; when the parametrization is adapted to the reflection symmetry of the slicing, βτ (0) = 0 must hold; by virtue of the S 1 -periodicity, βτ (±LS ) = ±LS for all τ . On each hypersurface Sτ , l is
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687
a coordinate, however, the 3-metric on Sτ is not of the form dl 2 + r 2 d2 for general ˙ µ ∂µ = (ϕ˙ + βϕ ˙ )∂σ + β∂ ˙ λ and µ ∂µ = ϕ β ∂σ + β ∂λ , where βτ . Now consider ,l β(τ, ·) = βτ (·). While the lapse α is given by (42), the shift vector field X i ∂i = x∂l , x = x(τ, l), depends on the gauge βτ . It is given explicitly by 1 ϕ ϕ˙ β˙ ϕ β˙ α + = − + , (44) β s2 − ϕ 2 β β s s2 − ϕ 2 β
where we again suppress the arguments τ, β(τ, l) on the r.h. side. Note that x(τ, 0) = 0 and x(τ, ±LS ) = 0, since ϕ (τ, ±LS ) = 0 and β(τ, ±LS ) = 0. We mention three gauges: when βτ (l) = l we recover the gauge used in Theorem 4.1, cf. (29), and x = −αs −1 (s 2 − ϕ 2 )−1/2 ϕ . Another interesting parametrization is the one given by β˙ = [s 2 − ϕ 2 ]−1 ϕ ϕ; ˙ this gives a vanishing shift, i.e. an “Eulerian” gauge. As a third possibility, the gauge given by β = [s 2 − ϕ 2 ]−1/2 suggests itself. Here the spatial metric reads dl 2 + r 2 d for all τ , as is easily seen by considering the pull-back of (27); ˙ we call this the “isotropic gauge”. The shift, which reads x = −αϕ + s 2 − ϕ 2 β, satisfies x = −α(K/3 − 2C/r 3 ), as can be shown by using (28) or by invoking the identity ∂τ (µ ,i ν ,j gµν ) = 2αKij + 2D(i Xj ) . But beware: the isotropic gauge is not consistent with the S 1 -periodicity requirement, and thus x(τ, ±LS ) = 0. The following relations are recorded for later use. Consider r = r((τ, l)) and t = t ((τ, l)) in any gauge. Restricted to S (i.e. for τ = 0 or σ = 0), we have ∂r ∂t KS r CS CS KS r −1 rα+ = − 2 α+xr , =V − 2 x . (45) ∂τ 3 r ∂τ 3 r x=−
To prove the first relation we compute r˙
τ =0
= (∂σ r)σ =0 ϕ˙ + ϕ β˙ τ =0 + (∂λ r)σ =0 β˙ τ =0 =
KS r CS , − 2 α+r x 3 r τ =0
where we have used (28) and the fact that ϕ(0, l) ≡ 0 and β(0, l) ≡ l. To show the second relation we note that the Killing vector is given by ξ = r ∂σ − [KS r/3 − CS /r 2 ]∂λ for σ = 0, cf. (12). Hence for ξν dx ν we obtain ξσ = −r and ξλ = −[KS r/3 − CS /r 2 ], both evaluated at σ = 0, cf. (27). Now, ∂ν t = −V −1 ξν , cf. (17), therefore KS r CS t˙τ =0 = (∂σ t)σ =0 α τ =0 + (∂λ t)σ =0 x τ =0 = V −1 r α + . − 2 x 3 r τ =0 (46) Proposition 4.6. Consider a spacetime KSSdS[T] that contains a CMC-slicing Sτ , τ ∈ (−τ¯ , τ¯ ), satisfying the properties (i) – (iii) of Theorem 4.1. This slicing uniquely corresponds to a smooth curve in KC0 , (−τ¯ , τ¯ ) τ → (K, C)(τ ) ∈ KC0 ,
(47)
˙ C)(τ ˙ ) of the curve is given such that (K, C)(0) = (KS , CS ). The tangent vector (K, through −
˙ )rmin (τ ) ˙ ) K(τ C(τ + 2 (τ ) αmin (τ ) , = rmin 3 rmin (τ )
(48)
(τ ) = r (τ ; 0) = (1/2)(dD(τ ; r)/dr)| where rmin (τ ) = r(τ ; 0), rmin r=rmin (τ ) , and αmin (τ ) = α(τ ; 0); α(τ ; l) is the lapse function of the slicing.
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Proof. Sτ is a compact CMC-hypersurface in KSSdS[T] for all τ , hence the CMCdata is characterized by (K(τ )), C(τ )) ∈ KC0 for all τ . Accordingly, Sτ generates a mapping (−τ¯ , τ¯ ) τ → (K, C)(τ ) = (K(τ ), C(τ )) ∈ KC0 ; clearly, (K, C)(0) = (K(0), C(0)) = (KS , CS ). By virtue of the uniqueness property of Sτ established in Theorem 4.1, the connection between the slicing Sτ and the map (K, C)(τ ) is one-toone. To prove that (K, C)(τ ) is a smooth curve in KC0 , consider Kij (τ ; r)K ij (τ ; r) =
K 2 (τ ) 6C 2 (τ ) + , 3 r6
(49)
cf. (6). Along the (smooth) slicing Sτ , Kij K ij and K are smooth functions, therefore C(τ ) and thus (K, C)(τ ) is smooth. Let α(τ ; l) be the lapse function of the slicing Sτ and X i ∂i = x(τ ; l)∂l the shift vector in an arbitrary gauge. Consider ∂τ [Kij (τ ; l)K ij (τ ; l)]. We have on one hand that 2 36C 2 12C (50) K K˙ + 6 C˙ − 7 r˙ , 3 r r where r˙ is given by (45). On the other hand, from the evolution equations and by employing (39), (Kij K ij )˙ =
12 C α 12 Cr α (−rD(r) + r − 3M) − 6 r r4 2 36C K 2C +2 + 3 K˙ − 7 r x . 3 r r
(Kij K ij )˙ = −
(51)
Equating (50) and (51), the terms involving the shift cancel, and we obtain r (τ ; l)α (τ ; l) − r (τ ; l)α(τ ; l) =
˙ )r(τ ; l) ˙ ) K(τ C(τ − 2 , 3 r (τ ; l)
from which Eq. (48) ensues by evaluation at l = 0.
(52)
Definition 4.7. Equation (48), i.e. −
˙ min Kr C˙ + 2 = rmin αmin , 3 rmin
(53)
together with K˙ > 0, defines a unique oriented direction field on KC0 . This is because , and α rmin , rmin min can be regarded as functions of (K, C) ∈ KC0 . (Note in particular that the lapse function is determined by Eq. (39), which only relies on the initial data sets.) Proposition 4.6 states that a slicing Sτ is uniquely represented by a (local) integral curve of the direction field. The following proposition turns this into a global statement: Proposition 4.8. Consider a spacetime KSSdS[T] that contains a CMC-slicing Sτ , τ ∈ (−τ¯ , τ¯ ), satisfying the properties (i)–(iii) of Theorem 4.1. This slicing can be maximally extended to a slicing Sτ , τ ∈ (τ− , τ+ ) ⊇ (−τ¯ , τ¯ ), satisfying (i)–(iii), such that (K, C)(τ ) → ∂(KC0 ) (τ → τ± ) .
(54)
Thus, the maximal extension of a slicing uniquely corresponds to a maximal integral curve of the oriented direction field in KC0 .
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689
Proof. Let (τ− , τ+ ) be the maximal interval of existence of the slicing Sτ provided by Theorem 4.1, and consider the associated curve (K, C)(τ ) in KC0 , which is an open piece of a maximal integral curve (K, C)mic (τ ) of the direction field defined on KC0 . Assume that (K, C)(τ ) → (K+ , C+ ) ∈ KC0 as τ → τ+ . Since Sτ is a hypersurface in KSSdS[T], T (τ ) = T for all τ ∈ (τ− , τ+ ). As a function of K and C on KC0 , T is continuous, cf. (26); it follows that T (K+ , C+ ) = limτ →τ+ T (K(τ ), C(τ )) = T. Thus, by Theorem 4.1, there exists a unique local slicing Sˆπ , π ∈ (−π¯ , π¯ ), in KSSdS[T], such that Sˆ0 is the CMC-hypersurface characterized by (K+ , C+ ). The slicing Sˆπ is represented by an open piece of the maximal integral curve (K, C)mic passing through (K+ , C+ ); hence Sτ ∪ Sˆπ is a slicing of KSSdS[T] that extends Sτ . This is a contradiction to the assumption. The argument for τ− is identical. 5. Properties of the Slicings, and Foliations In this section we prove that each spacetime KSSdS[T] contains a unique (maximally extended) slicing by compact CMC-hypersurfaces, and we show that this slicing is a foliation for a certain range of the time parameter. However, we begin with a discussion of the asymptotic behavior of slicings. Lemma 5.1. Consider the function rmax (K,C)
T (K, C) =
V
−1
(r) D
− 21
(K, C; r)
Kr C − 2 3 r
dr ,
(55)
rmin (K,C)
cf. (26). Let [0, 1) ν → (K, C)(ν) ∈ KC0 be a curve such that (K, C)(ν) → (K∂ , C∂ ) ∈ ∂(KC0 ) as ν → 1. Then √ (56a) T → +∞ if (K∂ , C∂ ) ∈ Ct ∪ (K = − 3) , √ (56b) T → −∞ if (K∂ , C∂ ) ∈ Cb ∪ (K = + 3) , √ √ √ √ where, however, the points ( 3, Ct ( 3)) and (− 3, Cb (− 3)) are excluded. √ √ Proof. Let (K∂ , C∂ ) ∈ Ct with K∂ ∈ (− 3, + 3). Choose > 0 small, and write rmin (ν)+
T (ν) =
V −1 √ D(ν; r)
rmin (ν)
K(ν)r C(ν) dr + − 2 3 r
rmax
(ν)
[. . . ] dr .
(57)
rmin (ν)+
The second integral converges to a constant as ν → 1, since rmin (ν) and rmax (ν) converge to the values of rmin and rmax at (K∂ , C∂ ), respectively, and the integrand converges uniformly. In r ∈ (rmin (ν), rmin (ν) + ) we make the expansion D (ν; rmin ) (r − rmin )2 + O((r − rmin )3 ) , (58) 2 where rmin = rmin (ν) and the prime denotes differentiation w.r.t. r in the present context. Then the first integral becomes √ (ν; r C 2 log[D )] Krmin min −1 (59) − 2 × − √ V (rmin ) 3 D (ν; rmin ) rmin (K∂ ,C∂ ) D(ν; r) = D (ν; rmin )(r − rmin ) +
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R. Beig, J. M. Heinzle
plus a constant plus a term of the order O(D (ν; rmin )). The first factor is a positive num2 is negative, cf. Fig. 10(a). With ν → 1, D (ν; r ber, since K∂ rmin /3−C∂ /rmin min (ν)) → 0, cf. Fig. 5(a); the equation D (rmin ) = 0 is in fact the defining equation for Cb,t , see Appendix A. Therefore, T (ν) diverges like − log[D (ν; rmin (ν))], i.e. T → √ ∞ as ν → 1. Analogously, we are able to prove that T = −∞ on Cb when K∂ = ± 3; 2 ); see Fig. 7. the different sign results from the different sign of (Krmin /3 − C/rmin √ √ √ let K∂ = ± 3, C∂ ∈ (Cb (± 3), Ct (± 3)). rmax (ν) diverges like √ Now 3/ − K(ν)2 /3 along the curve (K, C)(ν) as ν → 1, cf. Sect. 2. Choose r0 > rc such that D(ν; r) < 4, r − 2M > 0, and |K(ν)|r 3 > 6|C(ν)| for all r ∈ [r0 , rmax (ν)/2] for all ν sufficiently close to 1. Then,
r0 T − rmin
V −1 √ D
1 ≶ 4
C Kr − 2 3 r
rmax
/2
dr ≶ r0
rmax
/2
−
−1
r 2 3
Kr 3
V −1 √ D
dr = −
C Kr − 2 3 r
dr
rmax /2 1K log r , r0 4
(60)
r0
where we have suppressed the dependence on ν. We infer that T (ν) diverges √ at least 2 /3) as ν → 1; hence T → −∞ when K = like −(signK) log( − K(ν) 3 and ∂ √ T → ∞ when K∂ = − 3. A combination of√the arguments used for (59) and (60) √ also yields that √ T → −∞ √ when (K∂ , C∂ ) = ( 3, Cb ( 3)) and T → ∞ when (K∂ , C∂ ) → (− 3, Ct (− 3)). Proposition 5.2. Consider a spacetime KSSdS[T] that contains a CMC-slicing satisfying the properties (i)–(iii) of Theorem 4.1. Let Sτ , τ ∈ (τ− , τ+ ), be the maximal extension. Then √ √ (τ → τ+ ) , (61a) (K, C)(τ ) → ( 3, Ct ( 3)) √ √ (τ → τ− ) . (61b) (K, C)(τ ) → (− 3, Cb (− 3)) Hereby, the hypersurfaces Sτ converge to the asymptotic hypersurfaces S± , S+ in the future, S− in the past, √ 1 S± = r = √ 1 − 1 − 3 M ∪ (r = ∞) ⊆ KSSdS[T] (62)
−→ ←− Ct
T = −∞
T =∞
∞ T =
T =∞
K˙ > 0
C
K
− T =
∞
←− C b
−→
T = −∞
˙ C) ˙ satisfies K˙ > 0. On the boundaries T = ±∞ Fig. 7. On KC0 the direction field (K,
CMC-Slicings of KSSdS Cosmologies
691 r=∞
rb = r
rc =
rb = r
r
rc r
rc =
rb
=
rb = r
r
rb
rc
rb
=
=
=
=
rc
r
rc
r
r
r
=
r=∞
=
rc
rb
r
r
=
=
rb
r
r=0
r
r
=
=
r=0
r
rc
r=0
r=0
Fig. 8. The asymptotic hypersurfaces S± of the slicing Sτ
as τ → τ± , see Fig. 8. Remark 5.3. Proposition 5.2 is equivalent to the statement that all maximal integral √ curves√of the oriented direction √ field on KC √ 0 originate from the point (− 3, Cb (− 3)) and end in the point ( 3, Ct ( 3)). This is straightforward to prove: Proof. Each maximal integral curve (K, C)(τ ) of the direction field on KC0 is characterized by T (τ ) ≡ T = const, since the slicing is embedded in KSSdS[T]. Therefore, the limit set of the curve (K, C)(τ ) cannot a point on ∂(KC √ contain √ √ 0 ), where √ T = ±∞, which leaves only the two points (− 3, Cb (− 3)) and ( 3, Ct ( 3)) by Lemma 5.1. Together with K˙ > 0 this entails (61). To show the second part of the assertion, we recall that Sτ can be represented by t (τ ; r) in KSSdS[T], cf. (24). In analogy to the considerations in the proof of Lemma 5.1 we obtain rmin (τ )+
t (τ ; rmin (τ ) + ) =
√ rmin (τ )
V −1 D(τ ; r)
C(τ ) K(τ )r dr → ∞ − 2 3 r
(τ → τ+ ) (63)
for all > 0; moreover, rmin (τ ) → rmin (τ+ ) for τ → τ+ . Therefore, in the black hole region, Sτ converges to the hypersurface r = rmin (τ+ ) as τ → τ+ , where the convergence is uniform on each “cone” {t | t ∈ [−E, E] , E > 0}. From (23) we see that √ √ rmin (τ+ ) = (1− 1 − 3 M)/ . Similar considerations apply for the cosmological region; in particular, rmax (τ ) → ∞ as τ → τ+ . Hence the claim is established. Theorem 5.4. Let T ∈ R. The spacetime KSSdS[T] contains a unique (maximally extended) slicing Sτ , τ ∈ (τ− , τ+ ), such that a. Sτ is a compact CMC-hypersurface for all τ ∈ (τ− , τ+ ), b. Sτ is reflection symmetric for all τ . Along the slicing, c. K(τ ) is strictly monotonically increasing. Every slicing Sτ that satisfies (a) arises from Sτ by combining the flow of Sτ with an appropriate admixture of the Killing flow.
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K) ← Ct (
→ K= √
√ K = − 3
3
K) ← Cb (
→
˙ C) ˙ on KC0 for = 1, M = 1/4 Fig. 9. The direction field (K,
Remark 5.5. By definition, T is constant along the maximal integral curves of the oriented direction field on KC0 . Below we prove that, for each T, the equation T = T defines a unique integral curve, which corresponds to a unique slicing Sτ in KSSdS[T], τ ∈ (τ− , τ+ ), satisfying (a)–(c). The function T can thus be viewed as a “Hamilton function” for the oriented direction field on KC0 . Proof. Each maximal integral curve of the oriented direction field on KC0 has a unique point of intersection with the line K = const, since K˙ > 0 everywhere. Hence the pairs (0, C) with C ∈ (Cb (0), Ct (0)) parametrize the family of integral curves in KC0 . In the following we prove that ∂T (K, C)/∂C is positive for all (K, C) ∈ KC0 . By virtue of the asymptotic properties of T (K, C) established in Lemma 5.1, this implies that C → T (0, C) is a bijection between (Cb (0), Ct (0)) and R. Hence, there exists a unique pair (0, C) ∈ KC0 , such that T (0, C) = T, and thus a unique maximal integral curve in KC0 , such that T = T along the curve. Since the integral curve uniquely corresponds to a slicing Sτ satisfying (a)–(c), the claim of the theorem is established. We now show that ∂T (K, C)/∂C > 0 for all (K, C) ∈ KC0 . Consider the initial data generated by (KS , CS) ∈ KC0 and let again LS = L(KS , CS ), T = T (KS , CS ). The universal covering of the data is embedded as a Cauchy CMChypersurface S in the covering space KSSdS. In a neighborhood of S we introduce Gaussian coordinates (σ, λ), cf. the proof of Theorem 4.1. Now consider the (auxiliary) spacetime (−σ¯ , σ¯ ) × (−LS − ε, LS + ε) × S 2 , which we denote by KSSdS[T] ; it is an open subset of KSSdS and spatially incomplete. In KSSdS[T] the hypersurface S does not generate a unique slicing, but evolves into a one-parameter family of slicings; we are able to show this in a straightforward way by using the methods of the proof of Theorem 4.1. We define the mean curvature operator K as in (30) and an operator C, r 6 (ϕ(λ), λ) K[ϕ]2 C 2 [ϕ] := Kij [ϕ]Kij [ϕ] − , (64) 6 3 which assigns to every hypersurface σ = ϕ(λ) a function (−LS − ε, LS + ε) λ → C[ϕ](λ); for a CMC-hypersurface characterized by (K, C), C[ϕ] ≡ C. By construction,
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693
2 K[0] = KS and C[0] = CS . K and C are C 1 operators on Heven (−LS − ε, LS + ε) , K [0] = + a, cf. (35), and r3 − ∇i C [0] = 3
r3 3
∇i + b ,
(65)
where b(λ) = M + KC/3 − 4C 2 /r 3 and ∇ i (r 3 /3) ∇i = r 2 r ∂l . The set ker K [0] is not trivial, and neither is ker C [0]. However, the joint map (K , C )[0] is an isomorphism; see Appendix B. We may apply the inverse function theorem: there exists a unique continuously differentiable mapping (K, C)−1 such that (K, C)[(K, C)−1 ((κ, γ ))] = (κ, γ ) 0 0 . Hence, given a smooth for all (κ, γ ) of a neighborhood of (KS , CS ) in Heven × Heven function (−τ¯ , τ¯ ) τ → (K(τ ), C(τ )) such that (K(0), C(0)) = (KS , CS ), a unique slicing Sτ in KSSdS[T] is defined by using ϕτ := (K, C)−1 (K(τ ), C(τ )). By construction, the slicing Sτ in KSSdS[T] uniquely corresponds to a curve (K, C)(τ ) in a neighborhood of (KS , CS ) ∈ KC0 . Consider the hypersurface S and the slicing Sτ in KSSdS[T] such that S0 = S and K(τ ) = KS for all τ ; this slicing is represented by a curve (K, C)(τ ) = (KS , C(τ )), C(0) = CS , in KC0 . Let α(τ, l) be the lapse function of the slicing and x(τ, l) the shift vector in an arbitrary gauge. Making use of (45) we find that ∂ 2r ∂r (0, LS ) = (0, LS ) = ∂τ ∂l ∂τ
CS KS rmax − 2 3 rmax
α (0, LS ) + x(0, LS )rmax , (66)
= r (0, LS ). Hence, differentiation of the equation where rmax = r(0, LS ) and rmax r (τ, L(τ )) = 0 results in
α (0, LS ) KS rmax CS ˙ − x(0, LS ) . L(0) =− − 2 3 rmax rmax
(67)
We are now prepared to investigate the derivative of T (τ ) at τ = 0. The definition T (τ ) = t (τ, L(τ )) leads to ∂t ∂t α (0, LS ) ˙ T˙ (0) = (0, LS ) + (0, LS ) L(0) = , ∂τ ∂l rmax
(68)
where we have employed (24) and (45). By virtue of Eq. (53) (which holds also for r 2 α(0, 0), and thus ˙ slicings of KSSdS[T] ), we have C(0) = rmin min r α (0, LS ) α0 (LS ) rmin ∂T (K, C) 1 1 α0 (2LS ) = 2 2 min = , (69) = 2 r 2 r 2 r 2 (KS ,CS ) ∂C α(0, 0) rmax 2 rmin rmin rmin rmax min min min
where we have also used (100) from Appendix B; α0 denotes the solution of the homogeneous lapse equation (38), associated with (KS , CS ), with initial data α0 (0) = 1. By virtue of Lemma 4.4, α0 is not periodic, so that α0 (2LS ) is non-zero; moreover, since α0 (2LS ) = 0 holds irrespective of the choice of (K, C) = (KS , CS ), it has a definite sign for all (K, C) ∈ KC0 . Consequently, also ∂T (K, C)/∂C ≷ 0 for all (K, C). Indeed ∂T /∂C > 0 by virtue of the asymptotic behavior of T (K, C) described in Lemma 5.1. This establishes the claim of the theorem.
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R. Beig, J. M. Heinzle
Remark 5.6. The slicing Sτ in KSSdS[0], which constitutes the only time-symmetric spacetime in the family KSSdS[T], is represented by a distinguished integral curve in KC0 that is invariant under the inversion. This curve passes through the origin (0, 0) which is associated with a CMC-hypersurface of time-symmetry, i.e. the hypersurface t = 0, which connects the bifurcation 2-spheres. Remark 5.7. The approach we have taken to establish the main results displays a pronounced interplay between the geometric analysis and the analysis of the function T . We have chosen this approach because the direct investigation of this function seems much harder; e.g. it is not straightforward to show directly that it has no critical points, since methods like the ones applied in [3] fail. Theorem 5.8. In KSSdS[T] consider the unique maximally extended slicing Sτ , τ ∈ (τ− , τ+ ), satisfying the properties (a)–(c) of Theorem 5.4. There exists an interval (τ¯− , τ¯+ ) ⊆ (τ− , τ+ ), such that Sτ , τ ∈ (τ¯− , τ¯+ ), is a foliation. Proof. Consider the lapse equation (39) for an umbilical pair (K, C) ∈ KC0 , i.e. C = 0. In this case a = − K 2 /3 = const, and α
(K,C=0)
−1 K2 ˙ =K − = const > 0 3
(70)
is the unique even solution on the domain S 1 provided by Corollary 4.5. Let α(τ, l) be the lapse function of the slicing Sτ in KSSdS[T] which is represented by (K, C)(τ ) in KC0 . The asymptotic properties (61) imply that there exists τ0 such that C(τ0 ) = 0. At τ = τ0 , α is given by (70), i.e. α(τ0 , l) = const > 0. Since α(τ, l) continuously depends on τ , cf. (42), there exists an interval (τ¯− , τ¯+ ) τ0 , such that α(τ0 , l) > 0 for all τ ∈ (τ¯− , τ¯+ ). Hence Sτ is a foliation for τ ∈ (τ¯− , τ¯+ ). Remark 5.9. Since the lapse function is explicitly known for C = 0, the direction field ˙ C) ˙ at C = 0 can be computed explicitly as well. From (70) we obtain (K, −1 K2 ˙ ˙ C (K,C=0) = KM − = const > 0 . 3
(71)
We infer that every integral curve intersects the line C = 0 exactly once, hence the value of τ0 introduced in the proof of Theorem 5.8 is unique. In general, (τ¯− , τ¯+ ) = (τ− , τ+ ) in Theorem 5.8; this is proved in Appendix C. In particular, Corollary C.4 shows that (τ¯− , τ¯+ ) = (τ− , τ+ ) for all spacetimes KSSdS[T] with |T| large enough. Conjecture 5.10. There exist numbers 0 < Fc < F0 < 1 such that the following statement holds: if KSSdS[T] is a cosmological Kottler-Schwarzschild-de Sitter spacetime with 9M 2 ∈ [Fc , F0 ) and sufficiently small |T |, then the maximally extended slicing Sτ , τ ∈ (τ− , τ+ ), of Theorem (5.4) is a foliation. This conjecture is based on extensive numerical evidence, which we present in Appendix C.
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695
A. The Parameter Space KC0 of Compact CMC-Data A.1. Properties of KC0 . By definition, (K, C) ∈ KC iff there exists a neighborhood of (K, C) ∈ R2 such that all pairs of that neighborhood generate compact CMC-initial data. The function r(l) that determines the initial data is given by the equation r = ± D(r)
D(r) = D(K, C; r) = 1 −
with
¯ 2 r C2 2M¯ − + 4 , (72) r 3 r
¯ = − K 2 /3; the initial data is compact iff r(l) is where M¯ = M + CK/3 and periodic and thus interpretable as a function S 1 → R. Therefore, (K, C) ∈ KC iff the function D(r) possesses two positive (simple) zeros rmin and rmax , such√ that D(r) > 0 in (rmin , rmax ). (Accordingly, when viewed over r ∈ [rmin , rmax ], r = ± D(r) describes a closed curve, cf. Fig. 3(b), and r(l) becomes a periodic function which oscillates between rmin and rmax .) Assume C = 0; then D(r) → ∞ as r → 0, thus two positive extrema are necessary to obtain the desired profile of D(r), cf. Fig. 3(a). The critical points of D(r) are 3 8 3 8 3 3 ¯ 2 ¯ 2 , (73) (r+ ) = M¯ − M¯ 2 − C M¯ + M¯ 2 − C (r− ) = ¯ ¯ 3 3 2 2 which are positive, iff ¯ >0
,
For future reference we note that ¯ >0
⇔
M¯ > 0
,
C2 <
3 M¯ 2 . ¯ 8
√ √ K ∈ (− 3, + 3) .
(74)
(75)
When the conditions (74) are satisfied, r− (r+ ) is automatically a minimum (maximum) of D(r); however, the conditions are not sufficient to ensure the profile 3(a) of D(r). ¯ > 0. Note that the case C = 0 is simpler: D(r) possesses the desired profile iff 3 ¯ Proof: D(r) → −∞ (r → 0, r → ∞); r− does not exist; r+ = 3M/ is a maximum ¯ > 0. Moreover, D(r+ ) > 0, since 9M 2 ¯ ≤ 9M 2 < 1. iff We can effectively reduce the problem by one parameter by writing 2 ˜2 ˜ ˜ r ) = 1 − 2M − r˜ + C D(˜ r˜ 3 r˜ 4
with r˜ =
¯ , r
¯ , M˜ = M¯
¯ C˜ = C . (76)
˜ r ) has the desired profile if and only Investigating (76) we find that the function D(˜ ˜ ˜ if (M, C) lies in a certain connected open set whose boundaries are convex/concave ˜ C) ˜ and functions. Using these properties, via the variable transformation relating (M, (K, C) for given (M, ) we can prove KC = KC0 ∪ KC1 ∪ KC−1 ,
(77)
where the KCi are pairwise disjoint connected open sets; KC−1 arises from KC1 by inversion at the origin. KC0 is the connection component of (K, C) = (0, 0), it is invariant under the inversion, cf. Fig. 4. KC0 is enclosed by √ the functions Ct (K) and Cb (K) = −Ct (−K), and the vertical straight lines K = ± 3.
696
R. Beig, J. M. Heinzle
The functions Cb,t (K) are only known implicitly; the defining equation for Cb,t 3 + 3M ¯ = 2r− , where , ¯ and r− depend on ¯ − ¯ M, is D(r − ) = 0, or, equivalently, r ¯ up to terms of the order O( ¯ 2 ), we obtain the K, Ct,b (K) . In the limit of small , approximate solutions 2 √ √ 2 1 ∓ 3KM ±1 ∓ ¯ √ 3 3 Cb,t (K) = 2 ±1 ∓ 1 ∓ 3KM ±1 ± , √ K 2 K2 1 ∓ 3KM √ √ where the upper (lower) sign applies to Ct (Cb ). |Cb ( 3)| < |Ct ( 3)| follows. The functions Cb,t are strictly monotonically increasing. To show this we differentiate the defining equation D(r− ) = 0; we obtain ∂Cb,t 1 3 = r− . Cb,t ∂K 3
(78)
For the sake of completeness we note that Ct (K) is a convex function. A.2. rmin and rmax on KC0 . We investigate rmin and rmax as functions of (K, C) on KC0 . ¯ 2 /3, thus In the special case C = 0, D(r) = 1 − 2M/r − r 2 2 ξ +π ξ −π ¯ . rmin = √ cos , rmax = √ cos where cos ξ := 3M 3 3 ¯ ¯ (79) √ Two subcases deserve special attention: when K = 0 we have cos ξ = 3M and rmin , rmax coincide √ with the horizons of Schwarzschild-de Sitter, rmin = rb , rmax = rc . In the limit K → ± 3, (79) becomes √ 3 8 3 2 ¯ + O( ¯ ) ¯ . rmin = 2M + M , rmax = √ + O( ) (80) 3 ¯ Equation (79) shows that rmin ↓ and rmax ↑ when |K| ↑; hence, for C = 0, for all K, rmin ≤ rb and rmax ≥ rc . By recalling D(r) = V (r) + [Kr/3 − C/r 2 ]2 , we conclude from V (r) > 0 ∀r ∈ (rb , rc ) that rmin , rmax ∈ (rb , rc ) is excluded. Since rmin ≤ rb and rmax ≥ rc for C = 0, and since rmin and rmax are continuous functions of (K, C) on the connected domain KC0 , rmin ≤ rb and rmax ≥ rc must hold everywhere on KC0 . Suppose that rmin = rb for some (K, C) ∈ KC0 ; it then follows from D(rb ) = 0 that C = (rb3 /3)K. Conversely, consider (K, C) with C = (rb3 /3)K; from r 2 2M − + D(r) = 1 − r 3
Kr 3 Kr − 2b 3 3r
2 (81)
we obtain D(rb ) = 0 and dD/dr|rb > 0, therefore rmin = rb . Hence rmin (K, C) = rb
⇐⇒
C=
rb3 K. 3
(82)
CMC-Slicings of KSSdS Cosmologies
697
We conclude that there exists a straight line in KC0 , given by C = (rb3 /3)K, along which rmin attains the maximal √ possible value rb . This straight line intersects the boundary ∂(KC0 ) in the K = ± 3 vertical √ lines (and has √ no intersection with Cb,t ). To establish this result we verify that (rb3 /3) 3 < Ct ( 3) and we note that r− < rb everywhere, so that the slope of Ct (K), cf. (78), is always less than the slope of the straight line. Hence, this “line of maximal rmin ” divides KC0 into two regions, an upper 2 ] = 0 (left) half and a lower (right) half. In each of the two halves [Krmin /3 − C/rmin √ 2 holds, since [Krmin /3 − C/rmin ] = ± −V (rmin ); it follows from the connectedness of 2 ] < 0 in the upper the regions and the continuity of the function that [Krmin /3 − C/rmin 2 left half and [Krmin /3 − C/rmin ] > 0 in the lower right half, see Fig. 10(a). Analogously, we find a straight line of minimal rmax in KC0 , rmax (K, C) = rc
⇐⇒
C=
rc3 K. 3
(83)
The straight line of minimal rmax , C = (rc3 /3)K, intersects Cb,t (K) and so defines an 2 ] < 0 and a (lower) right half, where (upper) left half of KC0 , where [Krmax /3 − C/rmax 2 ] > 0 holds, see Fig. 10(b). [Krmax /3 − C/rmax Consider a curve (K, C)(ν) and regard rmin and rmax as functions of ν. By differentiating the equation D(rmin ) = 0 w.r.t. ν we obtain −1 ˙ min Krmin Kr dD C C˙ , (84) − 2 − 2 r˙min = −2 dr rmin 3 3 rmin rmin w.r.t. ν. and the analogous result for rmax ; the overdot denotes differentiation Consider the boundaries Cb,t as parametrized curves K(ν), Cb,t (K(ν)) and assume K˙ > 0. By definition, rmin = r− on Cb,t , whereby dD/dr = 0 at rmin , so that (84) is not applicable. However, by differentiating the equation dD/dr|rmin = 0 we are able to 3 /3)K, ˙ see (78), we obtain express r˙min in terms of regular expressions: using C˙ = (rmin −1 2 d D Krmin C along Cb,t . (85) K˙ − 2 r˙min = −2 dr 2 rmin 3 rmin
C
C
Krmin 3
Krmax 3
− 2C < 0 rmin
− 2C > 0 rmin
rmax
K
K
Krmin 3
− 2C > 0
Krmax 3
− 2C < 0 rmax
Fig. 10. rmin is maximal, rmin = rb , along a straight line. It divides KC0 in two halves, which are 2 ≷ 0. On the boundaries, r characterized by Krmin /3 − C/rmin min increases as indicated by the arrows. For the first subfigure = 1, M = 1/5. rmax is minimal, rmin = rc , along another straight line, which 2 defines the regions Krmax /3√− C/rmax ≷ 0. On the boundaries, rmax increases as indicated by the arrows; rmax → ∞ (K → ± 3). For the second subfigure = 1, M = 3/10
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R. Beig, J. M. Heinzle
2 ] < 0 on C and d 2 D/dr 2 | Since [Krmin /3 − C/rmin t rmin > 0 we observe that r˙min > 0 along Ct ; by virtue of the reflection symmetry, r˙min < 0 along Cb . √ From the monotonicity properties of rmin along Cb,t and the lines √ K = ± 3, √ we conclude that rmin assumes its global minimum at the point (K, C) = ( 3, Cb ( 3)) and at the reflected point, see Fig. 10(a). Using (78) in the rmax -analogue of (84) we get −1 ˙ 3 r− Krmax Krmax dD C r˙max =−2 along Cb,t . (86) − 2 1− 3 dr rmax 3 rmax 3 rmax
We infer the properties of rmax √ along Cb,t as depicted in Fig. 10(b). In contrast to rmin , rmax is unbounded as K → 3. ¯ we can calculate rmin and rmax along Cb,t up to any desired order of . ¯ For small ¯ → 0, Along Cb,t for √ √ √ ¯ (±1 ∓ 1 ∓ 3KM) √ 3 3 3M rmin = ± (87) ±1 ∓ 1 ∓ 3KM ±1+ √ K 4 K 1 ∓ 3KM ¯ 2 ), where the upper sign applies to Ct , the lower sign to Cb . modulo terms of order O( ¯ = 0, Eq. (87) becomes Setting √ √ √ 1 rmin = √ for K = 3, C = Cb,t ( 3). (88) ±1 ∓ 1 ∓ 3 M Finally, along Cb,t , rmax is given by 2 ξ¯ − π ¯ 3/2 ) rmax = √ cos + O( 3 ¯
¯ , cos ξ¯ := 3M¯
where
(89)
¯ in D(r). which follows when C 2 /r 4 is neglected M/r against √ −1 √ For a given r0 ∈ ( (1 − 1 − 3 M), rb ), the set of all (K, C) such that rmin (K, C) = r0 is (a segment of) a straight line in KC0 : D(r0 ) = 0 holds iff (K, C) is chosen according to r03 (90) K ∓ r02 −V (r0 ) ; 3 moreover, dD/dr|r0 > 0 holds for (a segment of ) the described straight line; hence rmin = r0 there. The lines of constant rmin are depicted in Fig. 11(a). Analogously, rmax is constant along straight lines (90), where r0 ∈ (rc , ∞); see Fig. 11(b). C=
C
C
K
K
Fig. 11. rmin and rmax are constant along straight lines in KC. We chose = 1 and M = 1/5, M = 3/10
CMC-Slicings of KSSdS Cosmologies
699
A.3. Hypersurfaces r = const. An alternative way of investigating the space KC is based on an analysis of the CMC-data associated with r = const hypersurfaces.√In KSSdS the hypersurfaces r = const are spacelike hypersurfaces with unit normal −V (r) ∂/∂r when r ∈ (0, rb )∪(rc , ∞).A hypersurface r = const possesses constant mean curvature, the induced CMC-data is represented by the constants K, C, 2 1 r K2 3M 2 K =∓ √ , C =±√ 1− − + r r 2 . (91) −2 + r 3 r −V (r) 3 √ The induced metric is dl 2 + r 2 d2 with l = t −V (r) ∈ (−∞, ∞); identifying l = l0 with l = l1 > l0 leads to compact CMC-data, however, no canonical choice of l0 , l1 exists. The r = const initial data sets play the role of borderline cases: r = const data arises when the function D(r) possesses a double zero. We infer that the pair (K, C) of (91) lies on the boundary of KC0 : √ √ √ (K, C) ∈ Cb,t when 1 − 1 − 3 M < r < −1 + 1 + 3 M . (92) Equations (91) thus constitute a parametric representation of the boundaries Cb,t of KC0 . B. The Lapse Equation We consider the equation α + aα = 0 ,
i.e.
α +
2 r α +aα = 0, r
(93)
where a(l) = − K 2 /3 − 6C 2 /r 6 (l); note that there do not exist pairs (K, C) ∈ KC0 such that a(l) is non-positive, which follows from algebraic computations based on the results of Appendix A. We seek the general solution on the domain R. Equation (93) is homogeneous with periodic, even and odd coefficient functions. The general solution of (93) is a superposition of two principal solutions with symmetry properties: – Let α0 (l) denote the function that solves (93) with the initial conditions α0 (0) = 1, α0 (0) = 0; it is even. – The function αξ (l) = r (l) is odd; it solves (93) with initial conditions αξ (0) = 0, . αξ (0) = rmin The function αξ is positive in (0, L) but negative in (−L, 0); αξ (0) = αξ (L) = 0; αξ is a periodic function. Geometrically speaking, αξ (l) = r (l) is the lapse function associated with the Killing vector ξ , cf. (11). To investigate the general solution of (93) we use Floquet’s theorem. Let us write (93) as a system of first order, α 0 1 α α = A(l) . (94) = α −a −2r /r α α The principal solution matrix is given by α0 (l) r (l)/rmin , (l, 0) = A(l)(l, 0) , (l, 0) = α0 (l) r (l)/rmin
(0, 0) =
10 01
. (95)
700
R. Beig, J. M. Heinzle
The Wronskian W (l, 0) = det (l, 0) satisfies the differential equation W (l, 0) = (trA(l)) W (l, 0), thus
W (l, 0) = det (l, 0) = exp 0
l
2 ˜ d l˜ = rmin trA(l) /r(l)2 .
(96)
From W (2L, 0) = 1 we see that the so-called monodromy matrix (2L, 0) fulfills 1 0 . (97) (2L, 0) = α0 (2L) 1 Floquet’s theorem states that (l, 0) = P (l, 0) exp(lQ(0)), where P (l, 0) is a periodic matrix and Q(0) is such that exp(2LQ(0)) = (2L, 0). Thus, 1 0 0 0 = P (l, 0) . (l, 0) = P (l, 0) exp l (lα0 (2L))/(2L) 1 α0 (2L)/(2L) 0 (98) It follows that ] αξ (l) α0 (2nL + l) = α0 (l) + n [α0 (2L)/rmin
∀l ,
∀n ∈ Z ,
(99)
in particular, α0 (2nL) = α0 (0) = 1. When we employ the symmetry properties of the coefficients in the differential equation (94) we are able to also establish a direct relation between the initial data at l = 0 and the solution at the half-period l = L: 2 rmin rmin 0 r 2 (L, 0) = rmax rmax r , where α0 (2L) = 2 α0 (L) min . (100) rmax α (L) max 0
rmin
We infer that α0 (l) (and thus the general solution of (93)) is periodic iff α0 (L) = 0. We prove α0 (L) = 0, which implies that the even solutions of (93) are not periodic, so that Lemma 4.4 is established. We analyze an explicit representation of α0 , which we obtain via an ansatz α0 (l) = β(l)r (l); we get ˙ α0 (l) = −Cr
l
1 dl + kr , r 2r 2
(101)
2 r , and k = const is chosen in such a way that the funcwhere C˙ is defined as C˙ = rmin min tion is even. The integral is divergent as l → 0 and l → ±L; however, by applying de /r )(r 2 /r 2 ), l’Hospital’s rule it is confirmed that α0 (0) = 1 and α0 (±L) = ±(rmin max min max cf. (100). When we view (the first half-period of) the function α0 as a function of r we can write r
1 1 d rˆ + k . (102) α0 (r) = C˙ D(r) − rˆ 2 D(ˆr )3/2
CMC-Slicings of KSSdS Cosmologies
701
We introduce δ(r) by defining δ(r) = r 4 D(r)(r − rmin )−1 (rmax − r)−1 ; the function δ(r) is positive in [rmin , rmax ]. Define
4r − 2(rmax + rmin ) dr , φ(r) = = √ 3/2 2 (r − rmin ) (rmax − r)3/2 (rmax − rmin ) (r − rmin )(rmax − r) (103) then α0 (r) becomes
α0 (r) = −C˙ D(r)φ(r)r 4 δ(r)−3/2 + C˙ D(r)
r
φ(r) r 4 δ(r)−3/2 dr ,
(104)
rmin
where the prime denotes differentiation w.r.t. r in the present context. Note that √ D(r)φ(r) is a regular bounded √ function so that the first term in (104) is regular and bounded. Using that dα0 /dl = D dα0 /dr we obtain by differentiating (104), α0 (L)
rmax C˙ = − D (rmin ) (r) r 4 δ(r)−3/2 dr , 2
(105)
rmin
√ where (r) = −4 (r − rmin )(rmax − r)(rmax − rmin )−2 is the integral of φ(r). Therefore, since (r) < 0 in (rmin , rmax ), to show that α0 (L) = 0 it is sufficient to show that the function r 4 δ(r)−3/2 is convex. 3 (r − r ), where ¯ The special case C = 0 is easy to treat. In this case, δ(r) = (/3)r neg rneg < 0, hence
r 4 δ(r)−3/2
∝
1 r (r − rneg )3
=
2 ) 3(r − rneg )4 (8r 2 − 4rneg r + rneg
4[r(r − rneg )3 ]5/2
>0, (106)
i.e. r 4 δ(r)−3/2 is convex. In the general case we show that the function r 3 δ(r)−1 is convex. Then, r 8/3 δ −1 = −1/3 r r 3 δ(r)−1 is convex, since r 3 δ(r)−1 is decreasing, and consequently r 4 δ(r)−3/2 = 8/3 (r δ −1 )3/2 is convex. An in-depth analysis of the properties of the zeros of δ(r) is essential to establish the claim. Asymptotically, for C → Cb,t , the zeros rmin and rmax of D(r) coincide to form the double zero r− , which is known explicitly, see (73). Hence δ(r) is known explicitly for C → Cb,t , δ(r) = − where
r0 (r0 − 3M)2 r , π 3 3r02 V (r0 ) r0
π(x) = 1 + 2x + 3x 2 + 2F x 3 + F x 4 with F = 1 +
(107a) 3r02 V (r0 ) , (107b) (r0 − 3M)2
and convexity of r 3 δ(r)−1 can be established; here, r0 is such that (K(r0 ), C(r0 )) describes a point on Cb,t via (91). Since δ(r) is a fourth-order polynomial, also its zeros are known explicitly for C → Cb,t . Combining this with an analysis of the variation of the zeros of δ as C varies, the claim can be established; the details are omitted here.
702
R. Beig, J. M. Heinzle
In order to differentiate (101), i.e. to be able to write down α0 (2L) in terms of quadratures, the integral must be regularized appropriately. The integral representation (101) of α0 can be “regularized” in several ways, e.g. α0 = 1 − r
l
1 r 2
2 rmin rmin 2
r
−r
(108a)
dl ,
0
r2 2 α0 = min + rmin r r2
l
2 2 dl − rmin r r3
0
l
1 r − r dl , r 2 r 2 min
(108b)
0
where the integrands are now regular at l = 0. When we differentiate the expression (108b), and manipulate the arising terms so that divergencies cancel, we obtain α0 rmax 2 rmin
1 = − 2 (rmax − r )(rmin − r ) + r r r
l
2 − r dl rmin + rmax 3 r
0
+r
l
r dl − r r 2r
0
l
1 (r − r )(rmin − r )dl , r 2 r 2 max
(109)
0
which can be evaluated at L to obtain α0 (L) and thus α0 (2L); the equation is mainly useful for numerical purposes. We consider now the lapse equation α + a α = K˙ ,
i.e.
α +
2 r α + a α = K˙ = const , r
(110)
where a(l) = − K 2 /3 − 6C 2 /r 6 (l), cf. (39). In first order form the system corre˙ Using the principal solution sponds to (94) with an additional inhomogeneity (0, K). matrix (95) of the homogeneous system, we obtain
l 0 α α = (l, 0) (0) + (l, s) ds α α K˙
(111)
0
by the method of variation of constants. Corollary 4.5 states that there exists a unique even periodic solution of (110); we give an alternative argument here. Equation (111) describes a periodic function if and only if (α, α )(2L) = (α, α )(0), i.e. iff
2L 0 α id − (2L, 0) (0) = (2L, s) ˙ ds , α K
(112)
0
where (2L, 0) is given by (97); (2L, s) can be computed easily, r (s) r (s) r(s)2 − 1 0 −1 rmin rmin (2L, s) = (2L, 0)(s, 0) = 2 . (113) rmin α0 (2L) 1 −α0 (s) α0 (s)
CMC-Slicings of KSSdS Cosmologies
703
Integration of the first component of (2L, s) (0, 1) yields
2L 0
r2 2 rmin
−
r
rmin
ds = −
r 3 2L 1 =0, 2 r 3 0 rmin min
(114)
i.e. the first component of Eq. (112) is satisfied identically. The second condition is a condition for α(0); it is fulfilled iff
2L K˙ 1 α(0) = − r(s)2 α0 (s)ds =: αmin . 2 α0 (2L) rmin
(115)
0
Equation (112) does not impose a condition on α (0); however, when α is required to be even, α (0) = 0 is necessary. Hence we have reproduced the result that there exists a unique even periodic solution of (110). The general solution of (110) is the linear combination αˆ = α + k0 α0 + kξ αξ , where k0 and kξ are constants; ] αξ (l) α(2nL ˆ + l) = α(l) + k0 α0 (l) + [kξ + k0 nα0 (2L)/rmin
∀l, ∀n ∈ Z (116)
follows from (99). When k0 = 0, kξ = 0, the solution is periodic but not even; when k0 = 0, kξ = 0, the solution is even but not periodic. To express αmin , cf. (115), in terms of quadratures we may use the regularization (108b). We obtain
L dl αmin 2 3 + =− L + 2rmax ˙ 3α (2L) r3 K 0 0
L −
3 − r 3 )(r − r ) (rmax min dl r 2r 2
0
−
3 − r3 1 rmin max . 3 rmin
By using (108a) and slightly different conventions we derive
rmax αmin = 0
⇔ rmin
r 2 dr 1 − √ D(r) 6
rmax
3 − r3 rmin D 3/2 (r)
2 rmax − D (r) dr = 0 , Dmax r2
rmin
(117) where D (r) = dD(r)/dr and Dmax = D (rmax ). We conclude this section by proving the claim made in the proof of Theorem 5.4, i.e. that the system
β + aβ = K˙ ,
3b 3C˙ r β − 3 β − 3 β = 3 , r r r
(118)
where a(l) = − K 2 /3 − 6C 2 /r 6 (l), and b(l) = M + KC/3 − 4C 2 /r 3 (l), has a ˙ It is unique even solution β(l) (on the domain R) for given even functions K˙ and C. straightforward to see that the system (118) is equivalent to the equation r β − r β =
˙ C˙ Kr − 2, 3 r
(119)
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R. Beig, J. M. Heinzle
which we have encountered already in (52). Since the coefficient r is odd, r even, there exists a unique solution β that is even; the general solution βˆ is a linear combination βˆ = β + const r . C. The Space KC0+ and Foliations Sτ In this section we discuss analytical and numerical results concerning the question of when a compact CMC-slicing Sτ in the spacetime KSSdS[T] is a foliation. These results strengthen the statement of Theorem 5.8. Let α(K, C; l) denote the unique even solution of the lapse equation (39) associated with (K, C) ∈ KC0 and a given constant K˙ > 0, cf. Corollary 4.5. We make the following Definition C.1. KC0+ is defined as the set of all (K, C) ∈ KC0 such that the associated lapse function α(K, C; l) is positive. Proposition C.2. There exists a neighborhood W of the line C = 0 in KC0 such that W ⊆ KC0+ .
(120)
Proof. The proof is similar to the proof of Theorem 5.8: αmin (K, C) and thus α(K, C; l) depends continuously on (K, C) ∈ KC0 , when K˙ is a given constant (or a continuous function on KC0 ), see Appendix B. Hence, since α(K, C; l) = const > 0 on the line C = 0, cf. (70), there exists a neighborhood W of C = 0 which is contained in KC0+ . Proposition C.3. There exists a neighborhood V (Cb ) of the curve (K, Cb (K)) | | K ∈ √ √ (− 3, 3) in KC0 and an analogous neighborhood V (Ct ) of the curve √ √ {(K, Ct (K)) | K ∈ (− 3, 3)} such that V (Ct ) ∩ KC0+ = ∅ . (121) V (Cb ) ∩ KC0+ = ∅ , √ √ √ √ There exists a neighborhood U ( 3) of {( √3, C) | C ∈ (Cb ( 3), Ct (√3))} in KC0 and an analogous neighborhood U (− 3) of the opposite line (− 3, C) | √ √ | C ∈ (Cb (− 3), Ct (− 3)) such that √ √ U (− 3) ∩ KC0 ⊆ KC0+ . (122) U ( 3) ∩ KC0 ⊆ KC0+ , Proof. In KSSdS[T] consider the foliation of the black hole region by r = const hypersurfaces. Recall that r = r0 = const is a CMC-hypersurface with metric dl 2 + r02 d2 and K = K0 , C = C0 given by (91). The lapse function αr of the r = const foliation at r = r0 is given by αr (l) ≡
K˙ , a(r0 )
where a(r0 ) = −
6C 2 K02 − 60 = const . 3 r0
(123)
˙ It is a solution of the lapse equation (39), where r = 0, i.e. αr solves αr + a(r0 )αr = K. We consider a hypersurface r = r such that r lies in the interval given in (92), so that 0 0 √ √ we have K0 ∈ (− 3, 3) and (without loss of generality) C0 = Ct (K0 ).
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Consider a neighborhood of (K0 , C0 ) in KC0 . Choose a pair (K, C) ∈ KC0 of that neighborhood and consider the associated CMC-hypersurface which is determined by the embedding t (r), see (24). For all small > 0, for all large E > 0, rmin (K,C)+
−1 V C Kr |t (rmin (K, C) + )| = − 2 dr > E, (124) √ 3 r D(K, C; r) rmin (K,C)
and |rmin (K, C) − r0 | < , provided that (K, C) is sufficiently close to (K0 , C0 ). This follows from (24) using the same techniques as in the proof of Lemma 5.1 and Proposition 5.2. In the subset t ∈ [−E, E] of the black hole region, the (monotonic) function r(t) thus satisfies |r(t) − r0 | + |∂r/∂t| ≤ 3, if (K, C) is sufficiently close to (K0 , C0 ), i.e., r(t) is approximately constant for t ∈ [−E, E]. Let, now, S be a compact CMC-hypersurface whose values (KS , CS ) are sufficiently close to (K0 , C0 ), i.e. (KS , CS ) − (K0 , C0 ) < δ; S can be approximated by the hypersurface r = r0 in a region t ∈ [−E, E] of the black hole. Consider the compact CMCslicing Sτ that contains the CMC-hypersurface S associated with (KS , CS ). Since the oriented direction field on KC0 is tangential to ∂(KC0 ) in the limit (K, C) → (K0 , C0 ), the integral curve (K, Sτ can be approximated (in at least the C 1 C)(τ ) that represents
sense) by the curve K(τ ), Ct (K(τ )) for all τ of some (small) τ -interval (independent of δ). It follows that the above statement carries over to the slicings Sτ : in some region t ∈ [−E, E] of the black hole, for all τ in some small interval, the slicing Sτ can be approximated by the slicing of r = const hypersurfaces through r = r0 . We conclude that K˙ αmin (K, C) → αr = (125) for (K, C) → (K0 , C0 ) . a(r0 ) Standard algebraic manipulations reveal that a(r0 ) < 0 for all (K0 , C0 ) ∈ Ct . Therefore, αmin (K, C) < 0 for all (K, C) sufficiently √ close √ to Ct , i.e. there exists a neighborhood V (Ct ) of the line {(K, Ct (K))|K ∈ (− 3, 3)} in KC0 such that V (Ct )∩KC0+ = ∅. The statement for Cb follows via the symmetry property of KC0 , hence the first claim is established. The proof of the second claim of the proposition follows the same principle: we √ consider a√CMC-hypersurface H in KSSdS with mean curvature K = 3 and 0 0 √ C0 ∈ (Cb ( 3), Ct ( 3)) of CMC-hypersur√ and we exploit the fact that the family √ √ faces Hτ generated by ( 3, C(τ )), where C(τ ) is running in (Cb ( 3), Ct ( 3)), forms a foliation of (a part of) KSSdS; in particular, α(r) > 0 for H0 . The slicing Hτ has been investigated in [6]. However, a proof of the positivity of the associated lapse function is missing. This gap can be closed via considerations similar to those of Appendix B, for a detailed discussion see [5]. When S is a √ compact CMC-hypersurface whose values (KS , CS ) are sufficiently close to (K0 = 3, C0 ), it √ can be suitably approximated by the hypersurface H0 , because rmin (K, C) → rmin ( 3, C0 ) and √ √ when (K, C) → ( 3, C0 ) (126) t (K, C; r) → t ( 3, C0 ; r) at least in C 1 , uniformly on every compact r-interval. Since √ the oriented direction field on KC0 is tangential to ∂(KC0 ) in the limit (K, C) → ( 3, C0 ) we infer √ √ for (K, C) → ( 3, C0 ) , (127) α(K, C; r) → α( 3, C0 ; r)
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R. Beig, J. M. Heinzle
√ therefore, if (K, C) is √in a sufficiently small neighborhood of ( 3, C0 ), then α(K, C; r) is positive since α( √ 3, C0 ; r) is positive. there exists a neighborhood √ √ Accordingly, √ √ U ( 3) of the line {( 3, C)|C ∈ (Cb ( 3), Ct ( 3))} in KC0 such that U ( 3)∩ KC0 ⊆ KC0+ . Corollary C.4. The compact CMC-slicing Sτ , τ ∈ (τ− , τ+ ), in the spacetime KSSdS[T], |T| sufficiently large, cannot be a foliation, i.e. (τ¯− , τ¯+ ) = (τ− , τ+ ) in Theorem 5.8. Proof. The integral curve (K, C)(τ ) associated with the slicing Sτ is characterized by T (τ ) ≡ T; when |T| is sufficiently large, it must pass through a given neighborhood of Cb or Ct (where T = ±∞, cf. Fig. 7) and thus through V (Cb ) or V (Ct ); in V (Cb ) and V (Ct ) the lapse function is not positive. Remark C.5. Let Sτ be an arbitrary slicing in KSSdS[T] that is not reflection symmetric and let αˆ denote its lapse function. Sτ arises from Sτ by combining the flow of Sτ with an appropriate admixture of the Killing flow, see Theorem 5.4, therefore α(l) ˆ = α(l) + kξ αξ (l) = α(l) + kξ r (l) for a constant kξ (depending on τ ), see Appendix B. Since r (l) is odd, αˆ > 0 whenever α > 0, i.e. if Sτ is not a foliation then there exists no other slicing Sτ in KSSdS[T] that is. Numerical investigations suggest that KC0+ is a connected set; the boundary ∂(KC √ √ 0+ ) consists of two smooth curves: one curve, P , connecting the points (− 3, C (− 3)) t √ √ √ √t and Ct ( 3)), and a second curve, Pb , connecting (− 3, Cb (− 3)) and √ ( 3,√ ( 3, Cb ( 3)). The curve Pt is given by Pt = {(K, C) | (αmin (K, C) = 0) ∧ (C > 0)} ,
(128)
cf. (117), Pb is the reflected curve, see Figs. 12 and 13. As with Cb,t we use the same symbols when we write the curves in parametric form, i.e. we also write C = Pb (K) and C = Pt (K). A slicing Sτ , τ ∈ (τ− , τ+ ), in KSSdS[T] is a foliation as long as the associated integral curve (K, C)(τ ) of the oriented direction field lies in KC0+ ; if the integral curve intersects Pt or Pb , Sτ cannot be a foliation for all τ . Since the integral curve is characterized by T (τ ) = T (K(τ ), C(τ )) ≡ T, at a possible intersection point (K, Pt (K)), T (K, Pt (K)) = T must hold, and analogously T (K, Pb (K)) = T for Pb . Therefore, the problem of whether or under what conditions a slicing is a foliation can be investigated by analyzing the function T (K, Pt (K)) and the equation T (K, Pt (K)) = T. Let Tf := inf K T (K, Pt (K));
√ 3
√ K = − 3
Pt (K )
KC0+ P b (K )
K=
→ ← Ct (K )
→ ← Cb (K )
Fig. 12. KC0+ and its boundaries Pt , Pb in the case = 1, M = 1/4
CMC-Slicings of KSSdS Cosmologies
707 1
0.95
√ − 3
√
3
Fig. 13. The depicted function represents Pt (K)/Ct (K)
i. if Tf > 0, then, for each T, the slicing Sτ in KSSdS[T] is not a foliation. ii. If Tf > 0, then slicings Sτ contained in spacetimes KSSdS[T] with |T| < Tf are foliations; slicings in KSSdS[T] with |T| ≥ Tf are not. To see (i) and (ii), we note that the equation T (K, Pt (K)) = T has a solution K iff T ≥ Tf . By virtue of the reflection symmetry in KC0 , T (K, Pb (K)) = −T (−K, Pt (−K)), thus the equation T (K, Pb (K)) = T is equivalent to the equation T (−K, Pt (−K)) = −T; it has a solution iff T ≤ −Tf . If Tf ≤ 0, then, for all T, at least one of the equations has a solution. If Tf > 0, then, for all |T| ≥ Tf , at least one of the equations has a solution; however, for all |T| < Tf , neither of the equations has a solution. This implies that the integral curve T (τ ) = T (with |T| < Tf ) in KC0 does neither intersect Pt nor Pb , but is entirely contained in KC0+ ; thus the associated slicing Sτ , τ ∈ (τ− , τ+ ), is a foliation. √ Lemma 5.1 entails that T (K, Pt (K)) → ∞ for K → − 3; moreover, the numerical results suggest that the function T (K, Pt (K)) is always strictly monotonically decreasing for K < 0. However, for K > 0, the properties of T (K, Pt (K)) depend on the chosen family of KSSdS[T]-spacetimes, i.e. on (, M)! In Fig. 14 we show the function T (K, Pt (K)) for several cases of M, where = 1. √ Numerical results suggest that the asymptotic behavior of T (K, Pt (K)) as K → 3 can be approximated by √ T (K, Pt (K)) = c1 + c2 log( 3 − K) , (129) where the constants c1 , c2 depend on (, M). This is consistent with the asymptotics of T obtained in the√proof of Lemma 5.1. The constant c2 is positive (so that T (K, Pt (K)) → −∞ as K → √3) for all M satisfying M < Mc ; c2 = 0 for M = Mc , and c2 < 0 for Mc < M < (3 )−1 , cf. Fig. 14. In the case M ≥ Mc , positivity of T (K, Pt (K)) is possible; indeed, there exists M0 such that Tf > 0 for all Mc ≤ M < M0 , see Figs. 14(d) and 14(e). Hence, for Mc ≤ M < M0 , the slicings Sτ in the spacetimes KSSdS[T] whose |T| is small enough, are foliations, cf. (ii). Thus there is strong numerical evidence that Conjecture 5.10 is true. From Fig. 14(c) we see that the equation T (K, Pt (K)) = T can have up to three solutions. Since in that case T (K, Pb (K)) = T has also one solution, an integral curve of the oriented direction field can switch between KC0+ and KC0 \KC0+ up to four times. The different scenarios can be read off from Fig. 14. We conclude this section by noting that the asymptotic behavior of Pt (K) is given by √ √ Pt (K) = Ct (K) 1 − [d1 + d2 ( 3 − K)]( 3 − K)2 , (130) where d1 and d2 are constants, d1 > 0, that depend on (M, ).
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R. Beig, J. M. Heinzle
T
T 1
1 0.5
√ 3
K
−1
√ 3 0.5
K
−3
−1
−5
(a) M = 0.100
(b) M = 0.200
T
T
1
1
K
0.5
0.5
(c) M = 0.215 T
1
1
(e) M = 0.250
K
√ 3
K
(d) M = 0.220
T
0.5
√ 3
√ 3
K
0.5
(f) M = 0.330
Fig. 14. The figures show the function √ T (K, Pt (K)) for K > 0 for several values of M; = 1 in T (K, P (K)) → −∞ (K → 3) for all M < Mc , i.e. for (a)–(c); T (K, Pt (K)) → +∞ all cases. t √ (K → 3) for all M > Mc , i.e. for (d)–(f); in the case M = Mc the function converges to a constant; Mc ≈ 0.218945. T (K, Pt (K)) ≥ const > 0 for all K when M ∈ [Mc , M0 ), i.e. for (d) and (e); M0 ≈ 0.268516
Acknowledgement. We would like to thank Christina Stanciulescu whose work [10] and handwritten notes were a very useful starting point for the present paper. J.M.H. would also like to thank Alan Rendall for helpful discussions.
References 1. Andersson, L.: The Global Existence Problem in General Relativity. In: P.T. Chru´sciel, H. Friedrich (eds.) The Einstein Equations and the Large Scale Behavior of Gravitational Fields, Basel-Boston: Birkh¨auser, 2004, pp. 71–120 2. Beig, R., Chru´sciel, P.T.: Killing initial data. Class. Quantum Grav. 14, A83–A92 (1997) ´ Murchadha, N.: Late time behavior of the maximal slicing of the Schwarzschild black 3. Beig, R., O hole. Phys. Rev. D 57(8), 4728–4737 (1998) 4. Chru´sciel, P.T., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mem. Soc. Math. France 94, 1–103 (2003)
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5. Heinzle, J.M.: CMC-Slicings of Kottler-Schwarzschild-de Sitter Cosmologies II. In preparation 6. Nakao, K.-i., Maeda, K.-i., Nakamura, T., Oohara, K.-i.: Constant-mean-curvature slicing of the Schwarzschild–de Sitter space-time. Phys. Rev. D 44, 1326–1329 (1991) ¨ 7. Kottler, F.: Uber die physikalischen Grundlagen der Einsteinschen Gravitationstheorie. Ann. Phys. (Germany) 56, 401–462 (1918) 8. Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations. Berlin-HeidelbergNew York: Springer, 2004 9. Rendall, A.D.: Constant mean curvature foliations in cosmological spacetimes. Helv. Phys. Acta 69, 490–500 (1996) 10. Stanciulescu, C.: Spherically Symmetric Solutions of the Vacuum Einstein Field Equations with Positive Cosmological Constant. Master’s thesis, University of Vienna, 1998 11. Valent, T.: Boundary Value Problems of Finite Elasticity. New York: Springer, 1987 Communicated by G.W. Gibbons
Commun. Math. Phys. 260, 711–725 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1432-4
Communications in
Mathematical Physics
Non-unitary minimal models, Bailey’s Lemma and N = 1, 2 Superconformal algebras Lipika Deka, Anne Schilling Department of Mathematics University of California One Shields Ave, Davis, CA 95616-8633 U.S.A. E-mail: [email protected]; [email protected] Received: 7 January 2005 / Accepted: 22 April 2005 Published Online: 20 September 2005 – © Springer-Verlag 2005
Abstract: Using the Bailey flow construction, we derive character identities for the N = 1 superconformal models SM(p , 2p + p ) and SM(p , 3p − 2p), and the N = 2 superconformal model with central charge c = 3(1 − 2p p ) from the nonunitary minimal models M(p, p ). A new Ramond sector character formula for representations of N = 2 superconformal algebras with central element c = 3(1 − 2p p ) is given. 1. Introduction Bailey’s lemma is a powerful method to prove q-series identities of the Rogers– Ramanujan-type [3]. One of the key features of Bailey’s lemma is its iterative structure which was first observed by Andrews [2] (see also [22]). This iterative structure called the Bailey chain makes it possible to start with one seed identity and derive an infinite family of identities from it. The Bailey chain has been generalized to the Bailey lattice [1] which yields a whole tree of identities from a single seed. The relevance of the Andrews–Bailey construction to physics was first revealed in the papers by Foda and Quano [14, 15] in which they derived identities for the Virasoro characters using Bailey’s lemma. By the application of Bailey’s lemma to polynomial versions of the character identity of one conformal field theory, one obtains character identities of another conformal field theory. This relation between the two conformal field theories is called Bailey flow. In [4] it was demonstrated that there is a Bailey flow from the minimal models M(p − 1, p) to N = 1 and N = 2 superconformal models. More precisely, it was shown that there is a Bailey flow from M(p − 1, p) to M(p, p + 1), and from M(p − 1, p) to the N = 1 superconformal model SM(p, p + 2) and the unitary N = 2 superconformal model with central charge c = 3(1 − p2 ). In the conclusions of [4] it was mentioned that this construction can also be carried out for the
Supported in part by NSF grant DMS-0200774.
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nonunitary minimal models M(p, p ) where p and p are relatively prime. In this paper we consider the nonunitary case. We show that starting with character identities for the nonunitary minimal model M(p, p ) of [6, 26], characters of the N = 1 superconformal models SM(p , 2p + p ), SM(p , 3p − 2p) and of the N = 2 superconformal model with central element c = 3(1 − 2p p ) can be obtained via the Bailey flow. We also give a new Ramond sector character formula for a representation of the N = 2 superconformal model with central element c = 3(1 − 2p p ). The character identities obtained from the Bailey flow construction are of BoseFermi type. The bosonic side is associated with the construction of singular vectors of the underlying conformal field theory. The fermionic side is usually manifestly positive and reflects the quasiparticle structure of the model. The paper is organized as follows. In Sect. 2 we provide the necessary background about Bailey pairs and fermionic formulas of the M(p, p ) models. This section is added to make this paper self-contained. For details the reader should consult [4, 6, 7]. In Sect. 3 the characters of the N = 1 supersymmetric models SM(2p + p , p ) and SM(3p − 2p, p ) are derived using the Bailey flow. Explicit fermionic expressions for these characters are given. In sect. 4 the background regarding N = 2 superconformal models is stated and a new character for the Ramond sector is derived. Then it is demonstrated how to obtain the characters of the N = 2 superconformal model with central element c = 3(1 − 2p p ) via the Bailey flow along with the explicit fermionic expressions for these characters. In Sect. 5 we conclude with some remarks. 2. Bailey’s Lemma In this section we summarize Bailey’s original lemma [2, 3] and the Bose-Fermi identities for the M(p, p ) minimal models [5, 6, 16, 26]. 2.1. Bilateral Bailey lemma. A pair (αn , βn ) of sequences {αn }n≥0 and {βn }n≥0 is called a Bailey pair with respect to a if βn =
n j =0
αj , (q)n−j (aq)n+j
(2.1)
where (a)n := (a; q)n =
n−1
(1 − aq k ),
k=0
1 . (1 − aq −k ) k=1
(a)−n := (a; q)−n = n
Following [4], we are going to use an extended definition in this paper called the bilateral Bailey pair. A pair (αn , βn ) of sequences {αn }n∈Z and {βn }n∈Z is said to be a bilateral Bailey pair with respect to a if βn =
n j =−∞
αj . (q)n−j (aq)n+j
(2.2)
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713
Theorem 2.1 (Bilateral Bailey lemma [2–4]). If (αn , βn ) is a bilateral Bailey pair then ∞
(ρ1 )n (ρ2 )n (aq/ρ1 ρ2 )n βn
n=−∞
=
∞ (aq/ρ1 )∞ (aq/ρ2 )∞ (ρ1 )n (ρ2 )n (aq/ρ1 ρ2 )n αn . (aq)∞ (aq/ρ1 ρ2 )∞ n=−∞ (aq/ρ1 )n (aq/ρ2 )n
(2.3)
This lemma has been used with various Bailey pairs and different specializations of the parameters ρ1 and ρ2 to prove many q-series identities (see for example [1, 4, 15, 25]). In this paper the bilateral Bailey lemma is used to derive character identities for N = 1, 2 superconformal algebras from nonunitary minimal models M(p, p ). A useful way to obtain new Bailey pairs from old ones is the construction of dual Bailey pairs. If (αn , βn ) is a bilateral Bailey pair with respect to a, the dual Bailey pair (An , Bn ) is defined as An (a, q) = a n q n αn (a −1 , q −1 ), 2
Bn (a, q) = a −n q −n
2 −n
βn (a −1 , q −1 ).
(2.4)
Then (An , Bn ) satisfies (2.2) with respect to a. 2.2. Bailey pairs from the minimal models M(p, p ). As shown by Foda and Quano [15], the Bose-Fermi character identities [5, 6, 16, 26] for the minimal models M(p, p ) are of the form Br(b),s (L, b; q) = q −Nr(b),s Fr(b),s (L, b; q), with Nr(b),s as given in [6] and ∞
L Br(b),s (L, b; q) = q 1 (L + s − b) − jp 2 j =−∞ L (jp−r)(jp −s) . −q 1 2 (L − s − b) + jp q Here
j (jpp +r(b)p −sp)
(q)n n = j q (q)j (q)n−j
(2.5) q
(2.6)
(2.7)
is the q-binomial coefficient. The function fermionic formula Fr(b),s (L, b; q) will be discussed in the next section. For simplicity we are going to write r for r(b). Following [15, 4] the identity (2.5) yields the bilateral Bailey pair relative to a = q b−s+2x , where x = L−2n−b+s , 2 j (jpp +rp −sp) if n = jp − x q αn = −q (jp−r)(jp −s) if n = jp − b − x , (2.8) 0 otherwise βn =
q −Nr,s (p,p ) Fr,s (2n + b − s + 2x, b; q). (aq)2n
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The dual Bailey pair to (2.8) relative to a = q b−s+2x is 2 j p (p −p)−jp (r−b)−j s(p −p)−x(b+x−s) q αˆ n = −q (jp −s)(j (p −p)+r−b)−x(b+x−s) 0 βˆn =
if n = jp − x if n = jp − b − x , otherwise
(2.9)
q Nr,s n n2 (p,p ) a q Fr,s (2n + b − s + 2x, b; q −1 ). (aq)2n
Inserting (2.8) and (2.9) into the bilateral Bailey lemma yields ∞
(ρ1 )n (ρ2 )n (aq/ρ1 ρ2 )n
n=0
=
(aq/ρ1 )∞ (aq/ρ2 )∞ (aq)∞ (aq/ρ1 ρ2 )∞
q −Nr(b),s (p,p ) Fr,s (2n + b − s + 2x, b; q) (aq)2n ∞ (ρ1 )jp −x (ρ2 )jp −x (aq/ρ1 ρ2 )jp −x (aq/ρ1 )jp −x (aq/ρ2 )jp −x
j =−∞
(ρ1 )jp −b−x (ρ2 )jp −b−x (aq/ρ1 )jp −b−x (aq/ρ2 )jp −b−x
×(aq/ρ1 ρ2 )jp −b−x q (jp−r)(jp −s)
×q j (jpp +rp −sp) −
(2.10)
and ∞
(ρ1 )n (ρ2 )n (aq/ρ1 ρ2 )n
n=0
=
(aq/ρ1 )∞ (aq/ρ2 )∞ (aq)∞ (aq/ρ1 ρ2 )∞
q Nr(b),s n n2 (p,p ) a q Fr,s (2n + b − s + 2x, b; q −1 ) (aq)2n ∞ (ρ1 )jp −x (ρ2 )jp −x (aq/ρ1 ρ2 )jp −x (aq/ρ1 )jp −x (aq/ρ2 )jp −x
j =−∞
(ρ1 )jp −b−x (ρ2 )jp −b−x (aq/ρ1 )jp −b−x (aq/ρ2 )jp −b−x
(2.11) ×(aq/ρ1 ρ2 )jp −b−x q (jp −s)(j (p −p)+r−b)−x(b+x−s) . ×q
j 2 p (p −p)−jp (r−b)−j s(p −p)−x(b+x−s)
−
As in [4], we are going to consider different specializations of the parameters ρ1 and ρ2 in (2.10) and (2.11) to get character identities for N = 1, 2 superconformal algebras. 2.3. Fermionic formulas for M(p, p ). So far we have only considered the bosonic side of (2.5) explicitly. It suffices for the purpose of this paper to state the fermionic formula for the case p < p < 2p with p and p relatively prime and r, s being pure Takahashi length. We follow [7, Sect. 4]. The fermionic formula depends on the continued fraction decomposition p = 1 + ν0 + p − p
1 ν1 +
.
1 ν2 + · · ·
1 νn0 + 2
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Define ti = i−1 j =0 νj for 1 ≤ i ≤ n0 + 1 and the fractional level incidence matrix IB and corresponding Cartan matrix B as for 1 ≤ j < tn0 +1 , j = ti δj,k+1 + δj,k−1 (IB )j,k = δj,k+1 + δj,k − δj,k−1 for j = ti , 1 ≤ i ≤ n0 − δνn0 ,0 , δ for j = tn0 +1 j,k+1 + δνn0 ,0 δj,k B = 2Itn0 +1 − IB , where In is the identity matrix of dimension n. Recursively define ym+1 = ym−1 + (νm + δm,0 + 2δm,n0 )ym , y m+1 = y m−1 + (νm + δm,0 + 2δm,n0 )y m ,
y−1 = 0, y −1 = −1,
y0 = 1, y 0 = 1.
Then the Takahashi length and truncated Takahashi length are given by j +1 = ym−1 + (j − tm )ym j +1 = y m−1 + (j − tm )y m
for tm < j ≤ tm+1 + δm,n0 with 0 ≤ m ≤ n0 .
For b = β+1 , r(b) = β+1 with tξ < β ≤ tξ +1 + δξ,n0 and s = σ +1 with tζ < σ ≤ tζ +1 + δζ,n0 the fermionic formula is given by (p,p ) Fr,s (L, b; q)
=q
kb,s
q
1 t 1 4 m Bm− 2 Au,v m
j =1
(mod 2)
m≡Qu,v
tn0 +1
nj + m j mj
q
,(2.12)
where kb,s is a normalization constant and n, m ∈ Ztn0 +1 such that n+m=
1 IB m + u + v + Le1 2
(2.13)
0 with ei the standard i th basis element of Ztn0 +1 , u = eβ − nk=ξ +1 etk , v = eσ − n0 k=ζ +1 etk and Qu,v , Au,v as defined in [7, Sect. 4.2]. The q-binomial is also defined for negative entries
n+m m
q
=
(q n+1 )m . (q)m
Note that
n+m m
q −1
=q
−nm
n+m m
q
.
(2.14)
In fact using (2.14) we get the following dual form of the fermionic formula that will be useful later on: (p,p )
Fr,s
(L, b; q −1 )
= q −kb,s
m≡Qu,v
1
t
1
1
1
t
q 4 m Bm− 2 Lm1 + 2 Au,v m− 2 m (u+v)
tn0 +1
j =1
nj + m j mj
q
. (2.15)
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L. Deka, A. Schilling
3. N = 1 Superconformal Character from M(p, p ) In this section we are going to consider the specialization in (2.10) and (2.11), ρ1 −→ ∞,
ρ2 = finite.
(3.1)
We will see that these give characters of the N = 1 superconformal model SM(p, p ) given by [10, 17], (p,p )
χ˜ r,s
(p,p )
(q) = χ˜ p−r,p −s (q)
∞ j (jpp +rp −sp) (jp−r)(jp −s) (−q r−s )∞ 2 2 = q , −q (q)∞
(3.2)
j =−∞
where 1 ≤ r ≤ p − 1, 1 ≤ s ≤ p − 1, p and (p − p)/2 are relatively prime and 1 if i is even (NS-sector), i = 2 (3.3) 1 if i is odd (R-sector). The central charge is c =
3 2
−
3(p−p )2 . pp
3.1. The model SM(p , 2p + p ). Specializing ρ1 −→ ∞ and ρ2 = −q x = 0 in (2.10) we find for b − s even (NS sector), (p ,2p+p ) χ˜ s,2r+b (q)
=
q 21 (n2 +nb−ns) (−q 21 )n+(b−s)/2 (q)2n+b−s
n≥0
(p,p )
q −Nr,s Fr,s
b−s+1 2
with
(2n + b − s, b; q), (3.4)
and for b − s odd (R-sector), (p ,2p+p ) χ˜ s,2r+b (q)
=
q 21 (n2 +nb−ns) (−q)n+(b−s−1)/2 (q)2n+b−s
n≥0
(p,p )
q −Nr,s Fr,s
(2n + b − s, b; q). (3.5)
To obtain an explicit fermionic formula set m0 = L = 2n + b − s and insert (2.12) into (3.4). Then using m0
1 2
(−q ) m0 = 2
2
q
1 m0 2 2 ( 2 −k)
m0 2
k
k=0
(3.6) q
we find m0
(p ,2p+p ) χ˜ s,2r+b (q)
=q
− 18 (b−s)2 −Nr,s +kb,s
∞ 2
1
1 m0 2 2 −k)
q 8 m0 + 2 ( 2
m0 =0 k=0 m≡Qu,v m0 even
×q
1 t 1 4 m Bm− 2 Au,v m
1 × (q)m0
m0 tn 0 +1 nj + m j 2 . (3.7) mj k q q j =1
Non-unitary minimal models, Bailey’s Lemma and N = 1, 2 Superconformal algebras
717
Setting p = (k, m0 , m) ∈ Ztn0 +1 +2 , (3.7) in the NS-sector can be rewritten as (p ,2p+p )
χ˜ s,2r+b
1 2 (q) = q − 8 (b−s) −Nr,s +kb,s
t
˜
1 t
1
˜
q 4 p Bp− 2 Au,v p +2
p∈Z n0 +1 ˜ u,v )i ,i≥2 pi ≡(Q
1
tn0 +1 +2
1 × (q)p2
j =1,j =2
˜ 2 (IB˜ p + u pj
+ v˜ )j
q
,
(3.8)
˜ where IB˜ = 2Itn0 +1 +2 − B,
2 −1 0 B˜ = −1 1 1 , 0 −1 B A˜ u,v u˜ t v˜ t t ˜ u,v Q
= = = =
(0, 0, Au,v ), (0, 0, ut ), (0, 0, vt ), (0, 0, Qtu,v ).
(3.9)
Similarly setting m0 = 2n + b − s in (3.5) and using m0 +1
(−q) m0 −1 2
m +1 2 0 m0 −1 1 m0 +1 1 ( −k)( −k) 2 2 = q2 2 , k 2 q
(3.10)
k=0
we get the fermionic formula in the R-sector, (p ,2p+p )
χ˜ s,2r+b
(q) =
1 − 1 ((b−s)2 +1)−Nr,s +kb,s q 8 2
1 × (q)p2
tn0 +1 +2
1
j =1,j =2
t
1 t
˜
1
˜
q 4 p Bp− 2 Au,v p +2
p∈Z n0 +1 ˜ u,v )i ,i≥2 pi ≡(Q
˜ 2 (IB˜ p + u pj
+ v˜ )j
q
,
(3.11)
˜ tu,v = (0, 1, Qtu,v ). ˜ A, ˜ v˜ are as in (3.9) and u˜ t = (1, 0, ut ), Q where B, 3.2. The model SM(p , 3p − 2p). Similarly using the same specialization with the dual Bailey pair in (2.11) we find for b − s even in the NS-sector, (p ,3p −2p)
χ˜ s,3b−2r
(q) =
q 3n2 (n+b−s) (−q 21 )n+(b−s)/2 n≥0
(q)2n+b−s
(p,p )
q Nr,s Fr,s
(2n + b − s, b; q −1 ), (3.12)
and for b − s odd in the R-sector,
718
L. Deka, A. Schilling
(p ,3p −2p) χ˜ s,3b−2r (q)
=
q 3n2 (n+b−s) (−q)n+(b−s−1)/2 n≥0
(q)2n+b−s
(p,p )
q Nr,s Fr,s
(2n + b − s, b; q −1 ). (3.13)
To obtain the fermionic formula, as before we are going to set m0 = 2n + b − s. Inserting (3.10) and (2.15) into (3.13) we get in the R-sector m0 +1
(p ,3p −2p) χ˜ s,3b−2r (q)
∞ 2 1 1 2 = q − 8 (3(b−s) +1)+Nr,s −kb,s 2
m0 =0 k=0 m≡Qu,v m0 odd
1
t
1
t
1
1
×q 2 (m0 +k −m0 k−m0 m1 ) q 4 m Bm− 2 m (u+v)+ 2 Au,v m m +1 tn 0 +1 0 1 nj + m j 2 × . mj k (q)m0 q q 2
2
(3.14)
j =1
Define p = (k, m0 , m) ∈ Ztn0 +1 +2 , so that (3.14) in the R-sector can be rewritten as (p ,3p −2p)
χ˜ s,3b−2r
(q) =
1 − 1 (3(b−s)2 +1)+Nr,s −kb,s q 8 2
1 × (q)p2
tn0 +1 +2
t
˜
1
˜
+2
p∈Z n0 +1 pi ≡(Q˜ u,v )i ,i≥2
1
j =1,j =2
1 t
q 4 p B p+ 2 Au,v p
˜ 2 (IB˜ p + u pj
+ v˜ )j
q
,
(3.15)
˜ u,v )t = (0, 1, Qtu,v ), and where IB˜ = 2Itn0 +1 +2 − B˜ , v˜ as in (3.9), u˜ t = (1, 0, ut ), (Q
2 −1 0 B˜ = −1 2 −1 , 0 −1 B A˜ u,v = (0, 0, Au,v − ut − vt ).
(3.16)
Similarly, for the NS-sector it follows from (3.12), (p ,3p −2p)
χ˜ s,3b−2r
3 2 (q) = q − 8 (b−s) +Nr,s −kb,s
t
˜
1 t
1
˜
q 4 p B p+ 2 Au,v p +2
p∈Z n0 +1 pi ≡(Q˜ u,v )i ,i≥2
1 × (q)p2
tn0 +1 +2
j =1,j =2
1
˜ 2 (IB˜ p + u pj
+ v˜ )j
(3.17) q
˜ u,v )t = (0, 0, Qtu,v ), u˜ t = (0, 0, ut ) and v˜ t = (0, 0, vt ). with B˜ and A˜ u,v as in (3.16), (Q
Non-unitary minimal models, Bailey’s Lemma and N = 1, 2 Superconformal algebras
719
4. N = 2 Character Formulas 4.1. N = 2 superconformal algebra and spectral flow. The N = 2 superconformal algebra A is the infinite dimensional Lie super algebra [13] with basis Ln , Tn , G± r ,C and (anti)-commutation relation given by [Lm , Ln ] = (m − n)Lm+n + 1 ± Lm , G± r = ( m − r)Gm+r , 2 [Lm , Tn ] = −nTm+n , 1 [Tm , Tn ] = cmδm+n,0 , 3 ± Tm , G± r = ±Gm+r ,
C 3 (m − m)δm+n,0 , 12
C 2 1 − {G+ (r − )δr+s,0 , r , Gs } = 2Lr+s + (r − s)Tr+s + 3 4 ± [Lm , C] = [Tn , C] = Gr , C = 0, + − − {G+ r , Gs } = {Gr , Gs } = 0,
where n, m ∈ Z, but r, s are integers in R-sector and half-integer in NS-sector. The element C is the central element and its eigenvalue c is parametrized as c = 3(1 − 2p p ), where p, p are relatively prime positive integers. It was observed in [18, 24] that there exits a family of outer automorphisms αη : A → A which maps the N = 2 superconformal algebras to itself. These are explicitly given by + ˆ+ αη (G+ r ) = Gr = Gr−η , − ˆ− αη (G− r ) = Gr = Gr+η ,
(4.1)
c αη (Ln ) = Lˆ n = Ln − ηTn + η2 δn,0 , 6 c αη (Tn ) = Tˆn = Tn − ηδn,0 . 3
This family of automorphisms is called spectral flow and η ∈ R is called the flow parameter. When η ∈ Z each sector of the algebra is mapped to itself. When η ∈ Z + 21 the Neveu-Schwarz sector is mapped to the Ramond sector and vice-versa. We are going to use the spectral flow η = ± 21 to map the NS-sector to the R-sector. 4.2. Spectral flow and characters. We denote the Verma module generated from a highest weight state |h, Q, c with L0 eigenvalue h, T0 eigenvalue Q and central charge c by Vh,Q . The character χVh,Q of a highest weight representation Vh,Q is defined as χVh,Q (q, z) = Tr Vh,Q (q L0 −c/24 zT0 ). Following [18] the character transforms under the spectral flow in the following way ˆ
ˆ
Tr Vh,Q (q L0 −c/24 zT0 ) = Tr Vhη ,Qη (q L0 −c/24 zT0 ),
(4.2)
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L. Deka, A. Schilling
where hη and Qη are the eigenvalues of Lˆ 0 and Tˆ0 , respectively, as defined in (4.1). This means the new character χVhη ,Qη (q, z) which is the trace of the transformed operators over the original representation equals the character of the representation defined by the eigenvalues hη and Qη of Lˆ 0 and Tˆ0 , respectively. So the new character is the character of the representation Vhη ,Qη . For η = 21 the spectral flow α 1 takes a NS-sector character to an R-sector character. 2
Let χVNS (q, z) be a NS-sector character corresponding to the representation Vh,Q . Then h,Q by (4.2) and (4.1) the new R-sector character χVRhη ,Qη (q, z) is derived using ˆ
ˆ
c
1
c
c
χVRhη ,Qη (q, z) = Tr Vh,Q (q L0 −c/24 zT0 ) = Tr Vh,Q (q L0 − 2 T0 + 24 − 24 zT0 − 6 ) c
c
c
1
T0
c
c
1
(q, zq − 2 ). (4.3) = q 24 z− 6 Tr Vh,Q (q L0 − 24 (zq − 2 ) ) = q 24 z− 6 χVNS h,Q 4.3. R-sector character from NS-sector character. To simplify notation we are going to use a slightly different notation for characters. Since we are only dealing with the NS (q, z). The vacuum character in the NS-sector for which h = 0, Q = 0, we write χˆ p,p R R-sector character is denoted by χˆ p,p (q, z) with the corresponding (h, Q) specified separately. Following [9, 12, 13, 18, 19] the vacuum character for the N = 2 superconformal algebra with central element c = 3(1 − 2p p ) in the NS-sector is given by NS χˆ p,p (q, z) 1 1 ∞ (1 + zq n− 2 )(1 + z−1 q n− 2 )
= q −c/24
(1 − q n )2
n=1 ∞
× 1− +
q p(n+1)(p (n+1)−1) +
∞
q
pn(p n+1)
+
1
zq p n(pn+1)−pn− 2
1
1 + zq p n− 2
n=1
1
1 + zq p n+ 2
n=0
1
zq p n(pn+1)+pn+ 2
+
1
z−1 q p n(pn+1)+pn+ 2
1
1 + z−1 q p n+ 2 1
z−1 q p n(pn+1)−pn− 2 . (4.4) + 1 1 + z−1 q p n− 2
This formula can be verified using the embedding diagram for the vacuum character as described in [13, 18] and can be rewritten as (as will be useful later) NS χˆ p,p (q, z)
=q
−c/24
1 1 ∞ (1 + zq n− 2 )(1 + z−1 q n− 2 ) (1 − q n )2
n=1 ∞
×
j =−∞
1 − q 2p j +1
q pj (p j +1)
1
1
(1 + zq p j + 2 )(1 + z−1 q p j + 2 )
.
(4.5)
The unitary case p = 1 of these character formulas was given in [11, 20, 21, 23]. In particular if we put z = 1 in (4.5) we obtain the following formula derived in [13]: NS χˆ p,p (q)
=q
−c/24
1 1 ∞ ∞ (1 + q n− 2 )2 pj (p j +1) 1 − q p j + 2 q . 1 (1 − q n )2 1 + qp j+ 2
n=1
j =−∞
(4.6)
Non-unitary minimal models, Bailey’s Lemma and N = 1, 2 Superconformal algebras
721
Let us apply (4.3) to the NS-sector vacuum character (4.5) to get a Ramond sector character. From (4.1) it follows that 1 c Lˆ 0 = L0 − T0 + , 2 24 c Tˆ0 = T0 − . 6 For the vacuum character in the NS-sector (h, Q) = (0, 0), so the new eigenvalues c are (hη , Qη ) = ( 24 , − 6c ) in the R-sector. Hence the new character in the R-sector η corresponds to (h , Qη ) and by (4.3), c
c
1
R − NS − ˆ p,p χˆ p,p (q, z) = q 24 z 6 χ (q, zq 2 )
(−z)∞ (−z−1 q)∞ (q)2∞ ∞ 1 − q 2p j +1 q pj (p j +1) . (1 + zq p j )(1 + z−1 q p j +1 ) j =−∞ c
= z− 6
4.4. N = 2 superconformal characters for c = 3(1 − (2.10) we obtain ∞
2p p ).
(4.7)
Using r = 0 and b = 1 in
q −N0,s (p,p ) F (2n + 1 − s + 2x, 1; q) (aq)2n 0,s n=0 ∞ (ρ1 )jp −x (ρ2 )jp −x (aq/ρ1 )∞ (aq/ρ2 )∞ = (aq/ρ1 ρ2 )jp −x (aq)∞ (aq/ρ1 ρ2 )∞ (aq/ρ1 )jp −x (aq/ρ2 )jp −x j =−∞
(ρ1 )jp −1−x (ρ2 )jp −1−x jp −1−x q jp(jp −s) . − (aq/ρ1 ρ2 ) (4.8) (aq/ρ1 )jp −1−x (aq/ρ2 )jp −1−x (ρ1 )n (ρ2 )n (aq/ρ1 ρ2 )n
In this section we consider the specialization ρ1 = finite, Taking the limit ∞ n=0
aq ρ1 ρ2
(ρ1 )n (ρ2 )n
ρ2 = finite.
−→ 1 in (4.8), we find
q −N0,s (p,p ) F (2n + 1 − s + 2x, 1; q) (aq)2n 0,s
∞ (ρ1 )∞ (ρ2 )∞ jp(jp −s) ρ1 ρ2 q 2(jp −x−1) − 1 = q . (4.9) (ρ1 ρ2 )∞ (q)∞ (1 − ρ1 q jp −x−1 )(1 − ρ2 q jp −x−1 )
j =−∞
722
L. Deka, A. Schilling 1
1
4.5. NS-sector characters. Let us set ρ1 = −zq x+ 2 , ρ2 = −z−1 q x+ 2 in (4.9), which implies a = q 2x and s = 1. Making the variable change j −→ −j in (4.9) and setting x = 0 we obtain ∞
1
1
(−zq 2 )n (−z−1 q 2 )n
n=0
=
q −N0,1 (p,p ) F (2n, 1; q) (q)2n 0,1
1 1 ∞ (−zq 2 )∞ (−z−1 q 2 )∞ jp(jp +1) 1 − q 2jp +1 q . (4.10) 1 1 (q)2∞ (1 + zq jp + 2 )(1 + z−1 q jp + 2 )
j =−∞
Comparing with (4.5), we obtain NS χˆ p,p (q, z)
=q
c − 24 −N0,1
1 1 ∞ (−zq 2 )n (−z−1 q 2 )n
(q)2n
n=0
(p,p )
F0,1
(2n, 1; q).
(4.11)
Setting z = 1 and inserting the fermionic formula (2.12), we find NS χˆ p,p (q)
=q
c − 24 −N0,1 +k1,1
1 ∞ (−q 2 )
(q)2n
n=0 tn0 +1
×
j =1
nj + m j mj
2 n
t
1
1
q 4 m Bm− 2 Au,v m
m≡Qu,v
.
q
(4.12)
Let us set m0 = 2n and use (3.6) to get m0
NS χˆ p,p (q) = q
×q
m0
∞ 2 2
c − 24 −N0,1 +k1,1
1 m0 2 1 m0 2 2 −k1 ) + 2 ( 2 −k2 )
q 2(
m0 =0 k1 =0 k2 =0 m≡Qu,v m0 even 1 t 1 4 m Bm− 2 Au,v m
1 (q)m0
m0 m0 tn 0 +1 nj + m j 2 2 . (4.13) mj k1 q k2 q q j =1
Define p = (k1 , k2 , m0 , m) ∈ Ztn0 +1 +3 , so that (4.13) can be rewritten as
c
NS − −N0,1 +k1,1 χˆ p,p (q) = q 24
t
1 t
1
ˆ
q 4 p Dp− 2 Au,v p +3
p∈Z n0 +1 ˆ u,v )i ,i≥3 pi ≡(Q
1 × (q)p3
tn0 +1 +3
j =1,j =3
1
ˆ 2 (ID p + u pj
+ vˆ )j
q
,
(4.14)
Non-unitary minimal models, Bailey’s Lemma and N = 1, 2 Superconformal algebras
723
where ID = 2Itn0 +1 +3 − D and
2 0 D= −1 0 Aˆ u,v uˆ t vˆ t ˆ tu,v Q
0 −1 2 −1 −1 1 0 −1
0 0 , 1 B
= (0, 0, 0, Au,v ), = (0, 0, 0, ut ), = (0, 0, 0, vt ),
(4.15)
= (0, 0, 0, Qtu,v ).
This gives a new fermionic expression for the NS-sector character.
4.6. Ramond sector characters. Let us set ρ1 = −zq x , ρ2 = −z−1 q x+1 in (4.9), which implies a = q 2x and s = 1. Setting x = 0 and changing j −→ −j we obtain ∞ (−z)n (−z−1 q)n n=0
(q)2n
(p,p )
q −N0,1 F0,1
(2n, 1; q)
∞ (−z)∞ (−z−1 q)∞ jp(jp +1) 1 − q 2jp +1 = q . (q)2 ∞ (1 + zq jp )(1 + z−1 q jp +1 )
(4.16)
j =−∞
Comparing with (4.7) we get R χˆ p,p (q, z)
=z
− 6c −N0,1
q
∞ (−z)n (−z−1 q)n
(q)2n
n=0
(p,p )
F0,1
(2n, 1; q).
(4.17)
Again using (2.12) in a similar way to the NS-sector and setting z = 1 we find R −N0,1 +k1,1 χˆ p,p (q) = 2q tn0 +1
×
j =1
∞ (−q)n−1 (−q)n
(q)2n
n=0
nj + m j mj
1
n k=0
1
m≡Qu,v
q
.
(4.18)
Using
(x)n =
t
q 4 m Bm− 2 Au,v m
1
(−x)(n−k) q 2 (n−k)(n−k−1)
n k q
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L. Deka, A. Schilling
and setting m0 = 2n, Eq. (4.18) can be rewritten as R χˆ p,p (q)
= 2q
−N0,1 +k1,1
m0 m0 2 −1 2
∞
1
q 4 (m0 +2k1 +2k2 −2m0 k1 −2m0 k2 ) 2
m0 =0 k1 =0 k2 =0 m≡Qu,v m0 even m0 1 t 1 1 1 − 1 m Bm− A m+ (k −k ) u,v 1 2 2 2 2 ×q 4 k1 (q)m0 q tn0 +1
j =1
nj + m j mj
2
2
m0 2
k2
q
q
.
(4.19)
Setting p = (k1 , k2 , m0 , m) ∈ Ztn0 +1 +3 this becomes 1 t 1 ˆ R −N0,1 +k1,1 q 4 p Dp− 2 Au,v p χˆ p,p (q) = 2q t
+3
p∈Z n0 +1 ˆ u,v )i ,i≥3 pi ≡(Q
×
1 (q)p3
tn0 +1 +3
j =1,j =3
1
ˆ 2 (ID p + u pj
+ vˆ )j
q
,
(4.20)
with the same notations as in (4.15) except Aˆ u,v = (1, −1, 0, Au,v ),
uˆ t = (−1, 0, 0, ut ),
vˆ t = (−1, 0, 0, vt ).
This gives a new fermionic expression of the new R-sector character. 5. Conclusion In this paper we only considered the vacuum character for the N = 2 superconformal algebra with central charge c = 3(1− 2p p ) with p < p in the NS-sector and the Ramond sector character derived from the vacuum character. We believe that similar Bailey flows exist for the general N = 2 superconformal characters, but explicit formulas are not yet available in the literature. The astute reader might have noticed that unlike in sect. 3 we did not carry out the Bailey flow for the dual Bailey pair in sect. 4, the reason being that the fermionic formula (p,p ) Fr,s (L, b; q) for p < p < 2p and r = b = 1 are not given in [6, 7]. A formula however does appear in [26]. The matrix D in this case is 2 0 −1 0 0 2 −1 0 D= . −1 −1 2 −1 0 0 −1 B Details will be available in [8]. Acknowledgements. Special thanks to Professor Gaberdiel and Hanno Klemm for their help through the jungle of literature regarding N = 2 character formulas and their help regarding the spectral flow of N = 2 superconformal algebras. We are grateful to both of them for their e-mail correspondences. We would also like to thank Professor Dobrev for helpful discussions.
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References 1. Agarwal, A.K., Andrews, G.E., Bressoud, D.M.: The Bailey lattice. J. Ind. Math. Soc. 51, 57–73 (1987) 2. Andrews, G.E.: Multiple series Rogers-Ramanujan type identities. Pac. J. Math. 114(2), 267–283 (1984) 3. Bailey, W.N.: Identities of Rogers-Ramanujan type. Proc. London Math. Soc.( 2(50), 1–10 (1949) 4. Berkovich, A., McCoy, B.M., Schilling, A.: N = 2 Supersymmetry and Bailey pairs. Physica A 228, 33–62 (1996) 5. Berkovich, A., McCoy, B.M.: Continued fraction and fermionic representations for characters of M(p, p ) minimal models, Lett. Math. Phys. 37, 49–66 (1996) 6. Berkovich, A., McCoy, B.M., Schilling, A.: Rogers-Schur-Ramanujan type identities for the M(p, p ) minimal models of conformal field theory. Commun. Math. Phys. 191, 325–395 (1998) 7. Berkovich, A., McCoy, B.M., Schilling, A., Warnaar, S.O.: Bailey flows and Bose-Fermi identities (1) (1) (1) for coset models (A1 )N × (A1 )N /(A1 )N+N . Nucl. Phys. B 499, 621–649 (1997) 8. Deka, L.: PhD thesis, in preparation 9. Dobrev, V.K.: Structure of Verma modules and characters of irreducible highest weight modules over N = 2 superconformal algebras. In: Clausthal 1986, Proceedings,Differential Geometric Methods in Theoretical Physics, H.D. Doebner, J.D. Hennig (eds.), Singapore: World Scientific, 1987, pp. 289–307 10. Dobrev, V.K.: Characters of the irreducible highest weight modules over the Virasoro and superVirasoro algebras. Rend. Circ. Mat. Palermo 14, 25–42 (1987) 11. Dobrev, V.K.: Characters of the unitarizable highest weight modules over the N = 2 superconformal algebras. Phys. Lett. B 186, 43–51 (1987) 12. D¨orrzapf, M.: The embedding structure of unitary N = 2 minimal models. Nucl. Phys. B 529, 639–655 (1998) 13. Eholzer, W., Gaberdiel, M.R.: Unitarity of rational N = 2 superconformal theories. Commun. Math. Phys. 186, 61–85 (1997) 14. Foda, O., Quano, Y.H.: Polynomial identities of the Rogers-Ramanujan type. Int. J. Mod. Phys. A 10, 2291–2315 (1995) 15. Foda, O., Quano,Y.H.:Virasoro charater identities from theAndrews-Bailey construction. Int. J. Mod. Phys. A 12, 1651–1675 (1996) 16. Foda, O., Welsh, T.A.: On the combinatorics of Forrester-Baxter models. In: Physical combinatorics (Kyoto, 1999), Progr. Math. 191, Boston, MA: Birkhuser Boston, 2000, pp. 49–103 17. Goddard, P., Kent, A., Olive, D.: Unitary representations of the Virasoro and super-Virasoro algebras. Comm. Math. Phys. 103 no. 1, 105–119 (1986) 18. Klemm, H.: Embedding diagrams of the N = 2 superconformal Algebra under spectral flow. Int. J. Mod. Phys. A19, 5263 (2004) 19. Klemm, H.: Private communication 20. Kiritsis, E.B.: Character formulae and the structure of the representations of the N = 1, N = 2 superconformal algebras. Int. J. Mod. Phys. A 3, 1871–1906 (1988) 21. Matsuo, Y.: Character formula of C < 1 unitary representation of N = 2 superconformal algebra. Prog. Theor. Phys. 77, 793–797 (1987) 22. Paule, P.: On identities of the Rogers–Ramanujan type. J. Math. Anal. Appl. 107, 255–284 (1985) 23. Ravanini, F., Yang, S.-K.: Modular invariance in N = 2 superconformal field theories. Phys. Lett. B 195, 202–208 (1987) 24. Schwimmer, A., Seiberg, N.: Comments on the N = 2, N = 3, N = 4 superconformal algebras in two dimensions. Phys. Lett. B 184, 191–196 (1987) 25. Slater, L.J.: A new proof of Rogers’s transformation of infinite series. Proc. London Math. Soc. (2) 53, 460–475 (1951) 26. Welsh, T.A.: Fermionic expressions for the minimal model Virasoro characters. In: Memoirs of the American Mathematical Society vd.F15, Providence, RI: Amer. Math. Soc., (2005) Communicated by L. Takhtajan
Commun. Math. Phys. 260, 727–762 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1291-z
Communications in
Mathematical Physics
Conformal Orbifold Theories and Braided Crossed G-Categories Michael Muger ¨ , Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands. E-mail: [email protected] Received: 5 April 2004 / Accepted: 28 September 2004 Published online: 12 February 2005 – © Springer-Verlag 2005
Dedicated to Detlev Buchholz on the occasion of his sixtieth birthday Abstract: The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G−LocA of twisted representations. This category is a braided crossed G-category in the sense of Turaev [60]. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT ; modular category ; 3-manifold invariant. Secondly, we study the relation between G−LocA and the braided (in the usual sense) representation category Rep AG of the orbifold theory AG . We prove the equivalence Rep AG (G − LocA)G , which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep AG . In the opposite direction we have G−LocA Rep AG S, where S ⊂ Rep AG is the full subcategory of representations of AG contained in the vacuum representation of A, and refers to the Galois extensions of braided tensor categories of [44, 48]. Under the assumptions that A is completely rational and G is finite we prove that A has g-twisted representations for every g ∈ G and that the sum over the squared dimensions of the simple g-twisted representations for fixed g equals dim Rep A. In the holomorphic case (where Rep A Vect C ) this allows to classify the possible categories G−LocA and to clarify the rˆole of the twisted quantum doubles D ω (G) in this context, as will be done in a sequel. We conclude with some remarks on non-holomorphic orbifolds and surprising counterexamples concerning permutation orbifolds.
Supported by NWO through the “pioneer” project no. 616.062.384 of N. P. Landsman. Department of Mathematics, Radboud University, Nijmegen, The Netherlands. E-mail: [email protected]
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1. Introduction It is generally accepted that a chiral conformal field theory (CFT) should have a braided tensor category of representations, cf. e.g. [41]. In order to turn this idea into rigorous mathematics one needs an axiomatic formulation of chiral CFTs and their representations, the most popular framework presently being the one of vertex operator algebras (VOAs), cf. [26]. It is, however, quite difficult to define a tensor product of representations of a VOA, let alone to construct a braiding. These difficulties do not arise in the operator algebraic approach to CFT, reviewed e.g. in [22]. (For the general setting see [24].) In the latter approach it has even been possible to give a model-independent proof of modularity (in the sense of [59]) of the representation category for a natural class of rational CFTs [29]. This class contains the SU (n) WZW-models and the Virasoro models for c < 1 and it is closed w.r.t. direct products, finite extensions and subtheories and coset constructions. Knowing modularity of Rep A for rational chiral CFTs is very satisfactory, since it provides a rigorous way of associating an invariant of 3-manifolds with the latter [59]. It should be mentioned that the strengths and weaknesses of the two axiomatic approaches are somewhat complementary. The operator algebraic approach has failed so far to reproduce all the insights concerning the conformal characters afforded by other approaches. (A promising step towards a fusion of the two axiomatic approaches has been taken in [61].) Given a quantum field theory (QFT) A, conformal or not, it is interesting to consider actions of a group G by global symmetries, i.e. by automorphisms commuting with the space-time symmetry. In this situation it is natural to study the relation between the categories Rep A and Rep AG , where AG is the G-fixed subtheory of A. In view of the connection with string theory, in which the fixpoint theory has a geometric interpretation, one usually speaks of ‘orbifold theories’. In fact, for a quantum field theory A in Minkowski space of d ≥ 2 + 1 dimensions and a certain category DH R(A) of representations [16] – admittedly too small to be physically realistic – the following have been shown [19]: (1) DH R(A) is symmetric monoidal, semisimple and rigid, (2) there exists a compact group G such that DH R(A) Rep G, (3) there exists a QFT F on which G acts by global symmetries and such that (4) F G ∼ = A, (5) the vacuum representation of F , restricted to A, contains all irreducible representations in DH R(A), (6) all intermediate theories A ⊂ B ⊂ F are of the form B = F H for some closed H G, and (7) DH R(F ) is trivial. All this should be understood as a Galois theory for quantum fields. These results cannot possibly hold in low-dimensional CFT for the simple reason that a non-trivial modular category is never symmetric. Turning to models with symmetry group G, we will see that G acts on the category Rep A and that Rep AG contains the G-fixed subcategory (Rep A)G as a full subcategory. (The objects of the latter are precisely the representations of AG that are contained in the restriction to AG of a representation of A.) Now it is known from models, cf. e.g. [11], that (Rep A)G Rep AG whenever G is non-trivial. This can be quantified as dim Rep AG = |G| dim(Rep A)G = |G|2 dim Rep A, cf. e.g. [64, 45]. Furthermore, it has been known at least since [11] that Rep AG is not determined completely by Rep A. This is true even in the simplest case, where Rep A is trivial but Rep AG depends on an additional piece of information pertaining to the ‘twisted representations’ of A. (Traditionally, cf. in particular [11, 12, 10], it is believed that this piece of information is an element of H 3 (G, T), but the situation is considerably more complicated as we indicate in Subsect. 4.2 and will be elaborated further in a sequel [49] to this work.
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Already this simplest case shows that a systematic approach is needed. It turns out that the right structure to use are the braided crossed G-categories recently introduced for the purposes of algebraic [7] and differential [60] topology. Roughly speaking, a crossed G-category is a tensor category carrying a G-grading ∂ (on the objects) and a compatible G-action γ . A braiding is a family of isomorphisms (cX,Y : X ⊗ Y → X Y ⊗ X), where X Y = γ (Y ), satisfying a suitably generalized form of the braid identities. In Sect. ∂X 2 we will show that a QFT on the line carrying a G-action defines a braided crossed G-category G−LocA whose degree zero part is Rep A. After some further preparation it will turn out that the additional information contained in G−LocA is precisely what is needed in order to compute Rep AG . On the one hand, it is easy to define a ‘restriction functor’ R : (G−LocA)G → RepAG , cf. Subsect. 3.1. On the other hand, the procedure of ‘α-induction’ from [35, 62, 4] provides a functor E : RepAG → (G−LocA)G that is inverse to R, proving the braided equivalence Rep AG (G−LocA)G .
(1.1)
Yet more can be said. We recall that given a semisimple rigid braided tensor category C over an algebraically closed field of characteristic zero and a full symmetric subcategory S that is even (all objects have twist +1 and thus there exists a compact group G such that S Rep G) there exists a tensor category C S together with a faithful tensor functor ι : C → C S. C S is braided if S is contained in the center Z2 (C) of C [5, 44] and a braided crossed G-category in general [48, 30]. Applying this to the full subcategory S ⊂ Rep AG of those representations that are contained in the vacuum representation of ι
F
A, we show that the functor E factors as E = (RepAG −→ RepAG S −→ G−LocA), where F : RepAG S → G−LocA is a full and faithful functor of braided crossed G-categories. For finite G we prove the latter to be an equivalence: G−LocA Rep AG S.
(1.2)
Thus the pair (Rep AG , S) contains the same information as G−LocA (with its structure as braided crossed G-category). We conclude that the categorical framework of [48] and the quantum field theoretical setting of Sect. 2 are closely related. In [29] it was proven that Rep A is a modular category [59] if A is completely rational. In Sect. 4 we use this result to prove that a completely rational theory carrying a finite symmetry G always admits g-twisted representations for every g ∈ G. This is an analogue of a similar result [13] for vertex operator algebras. (However, two issues must be noted. First, it is not yet known when a finite orbifold V G of a – suitably defined – rational VOA V is again rational, making it at present necessary to assume rationality of V G . Secondly, no full construction of a braided G-crossed category of twisted representations has been given in the VOA framework.) In fact we have the stronger result d(Xi )2 = d(Xi )2 =: dim LocA ∀g ∈ G, Xi ∈(G−LocA)g
Xi ∈LocA
where the summations run over the isoclasses of simple objects in the respective categories. Let us briefly mention some interesting related works. In the operator algebraic setting, conformal orbifold models were considered in particular in [64, 37, 27]. In [64] it is shown that AG is completely rational if A is completely rational and G is finite, a result that we will use. The other works consider orbifolds in affine models of CFT,
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giving a fairly complete analysis of Rep AG . The overlap with our model independent categorical analysis is small. Concerning the VOA setting we limit ourselves to mentioning [13, 14] where suitably defined twisted representations of A are considered and their existence is proven for all g ∈ G. Also holomorphic orbifolds are considered. The works [32, 30, 31] are predominantly concerned with categorical considerations, but the connection with VOAs and their orbifolds is outlined in [32, Sect. 5], a more detailed treatment being announced. [30, II] and [31] concern similar matters as [44, 48] from a somewhat different perspective. All in all it seems fair to say, however, that no complete proofs of analogues of our Theorems 2.21, 3.18 and 4.2 for VOAs have been published. The paper is organized as follows. In Sect. 2 we show that a chiral conformal field theory A carrying a G-action gives rise to a braided crossed G-category G−LocA of (twisted) representations. Even though the construction is a straightforward generalization of the procedure in the ungraded case, we give complete details in order to make the constructions accessible to readers who are unfamiliar with algebraic QFT. We first consider theories on the line, requiring only the minimal set of axioms necessary to define G−LocA. We then turn to theories on the circle, establish the connection between the two settings and review the results of [29] on completely rational theories. In Sect. 3 we study the relation between the category G−LocA and the representation category Rep AG of the orbifold theory AG , proving (1.1) and (1.2). In Sect. 4 we focus on completely rational CFTs [29] and finite groups, obtaining stronger results. We give a preliminary discussion of the ‘holomorphic’ case where Rep A is trivial. A complete analysis of this case is in preparation and will appear elsewhere [49]. We conclude with some comments and counterexamples concerning orbifolds of non-holomorphic models. Most results of this paper were announced in [45], which seems to be the first reference to point out the relevance of braided crossed G-categories in the context of orbifold CFT. 2. Braided Crossed G-Categories in Chiral CFT 2.1. QFT on R and twisted representations. In this subsection we consider QFTs living on the line R. We begin with some definitions. Let K be the set of intervals in R, i.e. the bounded connected open subsets of R. For I, J ∈ K we write I < J and I > J if I ⊂ (−∞, inf J ) or I ⊂ (sup J, +∞), respectively. We write I ⊥ = R − I . For any Hilbert space H, B(H) is the set of bounded linear operators on H, and for M ⊂ B(H) we write M ∗ = {x ∗ | x ∈ M} and M = {x ∈ B(H) | xy = yx ∀y ∈ M}. A von Neumann algebra (on H) is a set M ⊂ B(H) such that M = M ∗ = M
, thus in particular it is a unital ∗-algebra. A factor is a von Neumann algebra M such that Z(M) ≡ M ∩ M = C1. A factor M (on a separable Hilbert space) is of type III iff for every p = p 2 = p ∗ ∈ M there exists v ∈ M such that v ∗ v = 1, vv ∗ = p. If M, N are von Neumann algebras then M ∨ N is the smallest von Neumann algebra containing M ∪ N, in fact: M ∨ N = (M ∩ N ) . Definition 2.1. A QFT on R is a triple (H0 , A, ), usually simply denoted by A, where 1. H0 is a separable Hilbert space with a distinguished non-zero vector , 2. A is an assignment K I → A(I ) ⊂ B(H0 ), where A(I ) is a type III factor. These data are required to satisfy • Isotony: I ⊂ J ⇒ A(I ) ⊂ A(J ), • Locality: I ⊂ J ⊥ ⇒ A(I ) ⊂ A(J ) ,
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• Irreducibility: ∨I ∈K A(I ) = B(H0 ) (equivalently, ∩I ∈K A(I ) = C1), • Strong additivity: A(I ) ∨ A(J ) = A(K) whenever I, J ∈ K are adjacent, i.e. I ∩ J = {p}, and K = I ∪ J ∪ {p}, • Haag duality A(I ⊥ ) = A(I ) for all I ∈ K, where we have used the unital ∗-algebras A∞ = A(I ) ⊂ B(H0 ), I ∈K ⊥
A(I ) = Alg {A(J ), J ∈ K, I ∩ J = ∅} ⊂ A∞ . Remark 2.2. 1. Note that A∞ is the algebraic inductive limit, no closure is involved. We have Z(A∞ ) = C1 as a consequence of the fact that the A(I ) are factors. 2. The above axioms are designed to permit a rapid derivation of the desired categorical structure. In Subsect. 2.4 we will consider a set of axioms that is more natural from the mathematical as well as physical perspective. Our aim is now to associate a strict braided crossed G-category G − Loc A to any QFT on R equipped with a G-action on A in the sense of the following : Definition 2.3. Let (H0 , A, ) be a QFT on R. A topological group G acts on A if there is a strongly continuous unitary representation V : G → U(H0 ) such that 1. βg (A(I )) = A(I ) ∀g ∈ G, I ∈ K, where βg (x) = V (g)xV (g)∗ . 2. V (g) = . 3. If βg A(I ) = id for some I ∈ K then g = e. Remark 2.4. 1. Condition 3 will be crucial for the definition of the G-grading on G− Loc A. 2. In this section the topology of G is not taken into account. In Sect. 3 we will mostly be interested in finite groups, but we will also comment on infinite compact groups. The subsequent considerations are straightforward generalizations of the well known theory [16, 20, 21] for G = {e}. Since modifications of the latter are needed throughout – and also in the interest of the non-expert reader – we prefer to develop the case for non-trivial G from scratch. Readers who are unfamiliar with the following well-known result are encouraged to do the easy verifications. (We stick to the tradition of denoting the objects of End B by lower case Greek letters.) Definition/Proposition 2.5. Let B be a unital ∗-subalgebra of B(H). Let End B be the category whose objects ρ, σ, . . . are unital ∗-algebra homomorphisms from B into itself. With Hom(ρ, σ ) = {s ∈ B | sρ(x) = σ (x)s ∀x ∈ B}, t ◦ s = ts, s ∈ Hom(ρ, σ ), t ∈ Hom(σ, η), ρ ⊗ σ = ρ(σ (·)), s ⊗ t = sρ(t) = ρ (t)s, s ∈ Hom(ρ, ρ ), t ∈ Hom(σ, σ ), End B is a C-linear strict tensor category with unit 1 = idB and positive ∗-operation. We have End1 = Z(B). We now turn to the definition of G−Loc A as a full subcategory of End A∞ .
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Definition 2.6. Let I ∈ K, g ∈ G. An object ρ ∈ End A∞ is called g-localized in I if ρ(x) = x ∀J < I, x ∈ A(J ), ρ(x) = βg (x) ∀J > I, x ∈ A(J ). ρ is g-localized if it is g-localized in some I ∈ K. A g-localized ρ ∈ End A∞ is transportable if for every J ∈ K there exists ρ ∈ End A∞ , g-localized in J , such that ρ ∼ = ρ (in the sense of unitary equivalence in End A∞ ). Remark 2.7. 1. If ρ is g-localized in I and J ⊃ I then ρ is g-localized in J . 2. Direct sums of transportable morphisms are transportable. 3. If ρ is g-localized and h-localized then g = h. Proof. By 1, there exists I ∈ K such that ρ is g-localized in I and h-localized in I . If J > I then ρ A(J ) = βg = βh , and Condition 3 of Definition 2.3 implies g = h. Definition 2.8. G − Loc A is the full subcategory of End A∞ whose objects are finite direct sums of G-localized transportable objects of End A∞ . Thus ρ ∈ End A∞ is in G−Loc A iff there exists a finite set and, for all i ∈ , there exist gi ∈ G, ρi ∈ End A∞ gi -localized transportable, and vi ∈ Hom(ρi , ρ) such that vi∗ ◦ vj = δij and ρ=
vi ρi (·) vi∗ .
i
We say ρ ∈ G−Loc A is G-localized in I ∈ K if there exists a decomposition as above where all ρi are gi -localized in I and transportable and vi ∈ A(I ) ∀i. For g ∈ G, let (G−Loc A)g be the full subcategories of G−Loc A consisting of those ρ that are g-localized, and let (G−Loc A)hom be the union of the (G−Loc A)g , g ∈ G. We write Loc A = (G−Loc A)e . For g ∈ G define γg ∈ Aut(G−Loc A) by γg (ρ) = βg ρβg−1 , γg (s) = βg (s), s ∈ Hom(ρ, σ ) ⊂ A∞ . Definition 2.9. Let G be a (discrete) group. A strict crossed G-category is a strict tensor category D together with • a full tensor subcategory Dhom ⊂ D of homogeneous objects, • a map ∂ : Obj Dhom → G constant on isomorphism classes, • a homomorphism γ : G → Aut D (monoidal self-isomorphisms of D) such that 1. ∂(X ⊗ Y ) = ∂X ∂Y for all X, Y ∈ Dhom , 2. γg (Dh ) ⊂ Dghg −1 , where Dg ⊂ Dhom is the full subcategory ∂ −1 (g). If D is additive we require that every object of D be a direct sum of objects in Dhom . Proposition 2.10. G − Loc A is a C-linear crossed G-category with End1 = Cid1 , positive ∗-operation, direct sums and subobjects (i.e. orthogonal projections split).
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Proof. The categories (G−Loc A)g , g ∈ G are mutually disjoint by Remark 2.7.3. This allows to define the map ∂ : Obj (G−Loc A)hom → G required by Definition 2.9. If ρ is g-localized in I and σ is h-localized in J then ρ ⊗ σ = ρσ is gh-localized in any K ∈ K, K ⊃ I ∪ J . Thus G−Loc A is a tensor subcategory of End A∞ and Condition 1 of Definition 2.9 holds. By construction, G−Loc A is additive and every object is a finite direct sum of homogeneous objects. It is obvious that γg commutes with ◦ and with ⊗ on objects. Now, γg (s ⊗ t) = βg (s)βg (ρ(t)) = βg (s)γg (ρ)(βg (t)) = γg (s) ⊗ γg (t). Furthermore, if s ∈ Hom(ρ, σ ) then βg (s) βg ρ(x) = βg σ (x) βg (s), and replacing x → βg−1 (x) we find βg (s) ∈ Hom(γg (ρ), γg (σ )). Thus γg it is a strict monoidal automorphism of G−Loc A. Obviously, the map g → γg is a homomorphism. If ρ is h-localized in I and J > I then γg (ρ) A(J ) = βg ρβg−1 = βg βh βg−1 , thus γg (ρ) is ghg −1 -localized in I , thus Condition 2 of Definition 2.9 is verified. 1 = idA∞ is e-localized, thus in G − Loc A and End1 = Z(A∞ ) = Cid1 . Let p = p2 = p∗ ∈ End(ρ). There exists I ∈ K such that p ∈ A(I ), and by the type III property, cf. 2.24.1, we find v ∈ A(I ) such that vv ∗ = p, v ∗ v = 1. Defining ρ1 = v ∗ ρ(·)v we have v ∈ Hom(ρ1 , ρ), thus G−Loc A has subobjects. Finally, for any finite set and any I ∈ K we can find vi ∈ A(I ), i ∈ such that i vi vi∗ = 1, vi∗ vj = δij 1. If ρi ∈ G−Loc A we find that ρ = i vi ρi (·)vi∗ is a direct sum. Remark 2.11. Due to the fact that we consider only unital ρ ∈ EndA∞ , the category G − Loc A does not have zero objects, thus cannot be additive or abelian. This could be remedied by dropping the unitality condition, but we refrain from doing so since it would unnecessarily complicate the analysis without any real gains. 2.2. The braiding. Before we can construct a braiding for G−Loc A some preparations are needed. Lemma 2.12. If ρ is g-localized in I then ρ(A(I )) ⊂ A(I ) and ρ A(I ) is normal. Proof. Let J < I or J > I . We have either ρ A(J ) = id or ρ A(J ) = βg . In both cases ρ(A(J )) = A(J ), implying ρ(A(I ⊥ )) = A(I ⊥ ). Applying ρ to the equation [A(I ), A(I ⊥ )] = {0} expressing locality we obtain [ρ(A(I )), A(I ⊥ )] = {0}, or ρ(A(I )) ⊂ A(I ⊥ ) = A(I ), where we appealed to Haag duality on R. The last claim follows from the fact that every unital ∗-endomorphism of a type III factor with separable predual is automatically normal. Lemma 2.13. Let ρ, σ be g-localized in I . Then Hom(ρ, σ ) ⊂ A(I ). Proof. Let s ∈ Hom(ρ, σ ). Let J < I and x ∈ A(J ). Then sx = sρ(x) = σ (x)s = xs, thus s ∈ A(J ) . If J > I and x ∈ A(J ) we find sβg (x) = sρ(x) = σ (x)s = βg (x)s. Since βg (A(J )) = A(J ) we again have s ∈ A(J ) . Thus s ∈ A(I ⊥ ) = A(I ), by Haag duality on R. Lemma 2.14. Let ρi ∈ G−Loc A, i = 1, 2 be gi -localized in Ii , where I1 < I2 . Then ρ1 ⊗ ρ2 = γg1 (ρ2 ) ⊗ ρ1 .
(2.1)
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Proof. We have I1 = (a, b), I2 = (c, d), where b ≤ c. Let u < a, v > d and define K = (u, c), L = (b, v). For x ∈ A(K) we have ρ2 (x) = x and therefore ρ1 ρ2 (x) = ρ1 (x). By Lemma 2.12 we have ρ1 (x) ∈ A(K), and since γg1 (ρ2 ) is g1 g2 g1−1 -localized in I2 we find γg1 (ρ2 )(ρ1 (x)) = ρ1 (x). Thus (2.1) holds for x ∈ A(K). Consider now x ∈ A(L). By Lemma 2.12 we have ρ2 (x) ∈ A(L) and thus ρ1 ρ2 (x) = βg1 ρ2 (x). On the other hand, ρ1 (x) = βg1 (x) and therefore γg1 (ρ2 )ρ1 (x) = βg1 ρ2 βg−1 βg1 (x) = βg1 ρ2 (x), 1 thus (2.1) also holds for x ∈ A(L). By strong additivity, A(K) ∨ A(L) = A(u, v), and by local normality of ρ1 and ρ2 , (2.1) holds on A(u, v) whenever u < a, v > d, and therefore on all of A∞ . Remark 2.15. If one drops the assumption of strong additivity then instead of Lemma 2.12 one still has ρ(A(J )) ⊂ A(J ) for every J ⊃ I . Lemma 2.14 still holds provided I1 < I2 and I1 ∩ I2 = ∅. Recall that for homogeneous σ we write σρ = γ∂(σ ) (ρ) as in [60]. Definition 2.16. A braiding for a crossed G-category D is a family of isomorphisms cX,Y : X ⊗ Y → XY ⊗ X, defined for all X ∈ Dhom , Y ∈ D, such that (i) the diagram X⊗Y
s ⊗ t-
X ⊗ Y
cX ,Y cX,Y ? ?
X Y ⊗ X X - X Y ⊗ X t ⊗s
(2.2)
commutes for all s : X → X and t : Y → Y , (ii) for all X, Y ∈ Dhom , Z, T ∈ D we have cX,Z⊗T = idXZ ⊗ cX,T ◦ cX,Z ⊗ idT , cX⊗Y,Z = cX,YZ ⊗ idY ◦ idX ⊗ cY,Z ,
(2.3) (2.4)
(iii) for all X ∈ Dhom , Y ∈ D and k ∈ G we have γk (cX,Y ) = cγk (X),γk (Y ) .
(2.5)
Proposition 2.17. G−Loc A admits a unitary braiding c. If ρ1 , ρ2 are localized as in Lemma 2.14 then cρ1 ,ρ2 = idρ1 ⊗ρ2 = idρ1ρ2 ⊗ρ1 . Proof. Let ρ ∈ (G−Loc A)g , σ ∈ G−Loc A be G-localized in I, J ∈ K, respectively. Let I < J . By transportability we can find ρ ∈ (G−Loc A)g localized in Iand a unitary u ∈ Hom(ρ, ρ ). By Lemma 2.14 we have ρ ⊗ σ = γg (σ ) ⊗ ρ, thus the composite cρ,σ : ρ ⊗ σ
u ⊗ idσ
ρ ⊗ σ ≡ γg (σ ) ⊗ ρ
idγg (σ ) ⊗ u∗ - γg (σ ) ⊗ ρ
is unitary and a candidate for the braiding. As an element of A∞ , cρ,σ = γg (σ )(u∗ )u = βg σβg−1 (u∗ )u. In order to show that cρ,σ is independent of the choices involved pick
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ρ˜˜ ∈ (G−Loc A)g g-localized in I(we may assume the same localization interval since ˜˜ In view of ρ localized in I is also localized in I˜˜ ⊃ I) and a unitary u ∈ Hom(ρ, ρ). ∗ ∗ ˜˜ ρ Lemma 2.13 we have u u ∈ Hom(ρ, ) ⊂ A(I), implying γg (σ )(u u ) = u u∗ . The computation cρ,σ = γg (σ )(u∗ )u = γg (σ )(u∗ )(u u∗ )( uu∗ )u = γg (σ )(u∗ (u u∗ ))( uu∗ )u = γg (σ )( u∗ ) u = cρ,σ shows that cρ,σ is independent of the chosen ρ and u ∈ Hom(ρ, ρ ). Now consider σ, σ ∈ G − Loc A G-localized in J , ρ ∈ (G − Loc A)g and t ∈ Hom(σ, σ ). We pick I < J , ρ g-localized in Iand a unitary u ∈ Hom(ρ, ρ ). We define cρ,σ = γg (σ )(u∗ )u and cρ,σ = γg (σ )(u∗ )u as above. The computation cρ,σ ◦ idρ ⊗ t = γg (σ )(u∗ )u ρ(t) = βg σ βg−1 (u∗ )u ρ(t) = βg σ βg−1 (u∗ ) ρ (t)u = βg σ βg−1 (u∗ )βg (t)u
−1 ∗ = βg [σ βg (u )t]u = βg [tσβg−1 (u∗ )]u = βg (t)βg σβg−1 (u∗ )u = βg (t)γg (σ )(u∗ )u = βg (t) ⊗ idρ ◦ cρ,σ proves naturality (2.2) of cρ,σ w.r.t. σ . (In the fourth step ρ (t) = βg (t) is due to t ∈ Hom(σ, σ ) ⊂ A(J ), cf. Lemma 2.13, and the fact that ρ is g-localized in I < J .) Next, let ρ, ρ ∈ (G−Loc A)g , s ∈ Hom(ρ, ρ ) and let σ ∈ G−Loc A be G-localized in J . Pick I < J , ρ , ρ g-localized in Iand unitaries u ∈ Hom(ρ, ρ ), u ∈ Hom(ρ , ρ ). Then
cρ ,σ ◦ s ⊗ idσ = γg (σ )(u ∗ )u s = γg (σ )(u ∗ )(u su∗ )u
∗
∗ γg (σ )(su∗ )u = γg (σ )(u (u su ))u = ∗ = γg (σ )(s)γg (σ )(u )u = idγg (σ ) ⊗ s ◦ cρ,σ proves naturality of cρ,σ w.r.t. ρ. (Here we used the fact that ρ , ρ are g-localized in
∗
I, implying u su ∈ Hom( ρ, ρ ) ⊂ A(I) by Lemma 2.13 and finally γg (σ )(u su∗ ) =
∗ u su .) Next, let ρ ∈ (G−Loc A)g and let σ, η ∈ G−Loc A be G-localized in J . We pick ρ g-localized in I < J and a unitary u ∈ Hom(ρ, ρ ). Then cρ,σ ⊗η = γg (σ η)(u∗ )u = γg (σ η)(u∗ ) γg (σ )(u) γg (σ )(u∗ ) u = γg (σ )[γg (η)(u∗ )u]γg (σ )(u∗ )u = idγg (σ ) ⊗ cρ,η ◦ cρ,σ ⊗ idη proves the braid relation (2.3). Finally, let ρ ∈ (G−Loc A)g , σ ∈ (G−Loc A)h and let η ∈ G−Loc A be G-localized in J . Pick ρ ∈ (G−Loc A)g , σ ∈ (G−Loc A)h G-localized in I < J and unitaries
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M. M¨uger
∈ Hom(ρσ, ρ u ∈ Hom(ρ, ρ ), v ∈ Hom(σ, σ ). Then w = uρ(v) = ρ(v)u σ ), thus cρ⊗σ,η = γgh (η)(w ∗ )w = γgh (η)(u∗ ρ (v ∗ ))uρ(v) ∗ = γgh (η)(u )γgh (η) ρ (v ∗ )uρ(v) = γgh (η)(u∗ ) ρ γh (η)(v ∗ )uρ(v) ∗ = γgh (η)(u )u ρ[γh (η)(v ∗ )v] = γg (γh (η))(u∗ )u ρ[γh (η)(v ∗ )v] = cρ,γh (η) ⊗ idσ ◦ idρ ⊗ cσ,η , ρ , cf. Lemma 2.14, proves (2.4). The last claim follows where we used ρ γh (η) = γgh (η) from ρ1 ρ2 = ρ2ρ1 ρ2 , cf. Lemma 2.14 and the fact that we may take ρ = ρ and u = idρ in the definition of cρ,σ . It remains to show the covariance (2.5) of the braiding. Recall that cρ,σ ∈ Hom(ρ ⊗ σ, γg (σ ) ⊗ ρ) was defined as idγg (σ ) ⊗ u∗ ◦ u ⊗ idσ for suitable u. Applying the functor γk we obtain idγkgk−1 (γk (σ )) ⊗ γk (u)∗ ◦ γk (u) ⊗ idγk (σ ) ∈ Hom(γk (ρ) ⊗ γk (σ ), γkgk −1 (γk (σ )) ⊗ γk (ρ)), ρ )). Since this is of the same form as cγk (ρ),γk (σ ) and where γk (u) ∈ Hom(γk (ρ), γk ( since the braiding is independent of the choice of the intertwiner u, (2.5) follows. 2.3. Semisimplicity and rigidity. In view of Lemma 2.12 we can define Definition 2.18. G−Locf A is the full tensor subcategory of G−Loc A of those objects ρ satisfying [A(I ) : ρ(A(I ))] < ∞ whenever ρ is g-localized in I . The following is proven by an adaptation of the approach of [23]. Proposition 2.19. G − Locf A is semisimple (in the sense that every object is a finite direct sum of (absolutely) simple objects). Every object of G−Locf A has a conjugate in the sense of [36] and G−Locf A is spherical [3]. Proof. By standard subfactor theory, [M : ρ(M)] < ∞ implies that the von Neumann algebra M ∩ ρ(M) = End ρ is finite dimensional, thus a multi matrix algebra. This implies semisimplicity since G−Locf A has direct sums and subobjects. Clearly, it is sufficient to show that simple objects have conjugates, thus we consider ρ ∈ (G−Locf A)g g-localized in I . By the Reeh-Schlieder property 2.24.3, cf. e.g. [22], the vacuum is cyclic and separating for every A(I ), I ∈ K, giving rise to antilinear involutions JI = J(A(I ),) on H0 , the modular conjugations. Conditions 1-2 in Definition 2.3 imply V (g)JI = JI V (g) for all I ∈ K, g ∈ G. For z ∈ R and K = (z, ∞) it is known [23, 22] that jK : x → JK xJK maps A(I ) onto A(rz I ), where rz : R → R is the reflection about z. Thus jK is an antilinear involutive automorphism of A∞ . Choosing z to be in the right hand complement of I , the geometry is as follows:
I
z
rz I
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Let ρ be g-localized in rz I and u ∈ Hom( ρ , ρ) unitary. Dropping the subscript z and defining ρ = jρ jβg−1 ∈ EndA∞ , it is clear that ρ is g −1 -localized in I . It is easy to see that d(ρ) = d(ρ) and that ρ is transportable, thus in (G−Locf A)g − 1 . Now consider the subalgebras A1 = A(I ), A2 = A(I ) I ∈K I ⊂(−∞,z)
I ∈K I ⊂(z,∞)
A1 = id and ρ A2 = βg = of A∞ . We have A 1 = A
2 = J A
1 J . In view of ρ Ad V (g) we have ρ A1 = u ρ (·) u∗ = u · u∗ , ∗ ρ A2 = u ρ(·) u = u∗ βg (·) u = u∗ V (g) · V (g)∗ u. We therefore find ρρ A1 = ρj ρ jβg−1 A1 = Ad uJ u∗ V (g) J V (g)∗ = Ad uJ u∗ J , where we used the commutativity of J and V (g). Since the above expressions for ρ A1 , ρ A2 , ρρ A1 are ultraweakly continuous they uniquely extend to the weak closures A
1 , A
2 , A
1 , respectively. Now, (A1 )
J u∗ = uJ A
1 J u∗ = u A
2 u∗ uJ u∗ ρ(A1 )
uJ u∗ = uJ ρ = (u A 2 u∗ ) = (u A
1 u∗ ) = ρ(A
1 ) = ρ(A1 ) . Thus, J = uJ u∗ is an antiunitary involution whose adjoint action maps ρ(A1 )
onto ρ(A1 ) . Furthermore, u is cyclic and separating for ρ(A1 )
and we have (uJ u∗ )(u) = uJ = u and (ρ(x)Jρ(x)Ju, u) = (xJ xJ , ) ≥ 0 ∀x ∈ A
1 . Thus J is [28, Exercise 9.6.52] the modular conjugation corresponding to the pair (ρ(A1 )
, u), and therefore x → ρρ(x) = J(ρ(A1 )
,u) J(A
1 ,) x J(A
1 ,) J(ρ(A1 )
,u) is a canonical endomorphism γ : A
1 → ρ(A1 )
[34]. Since [A
1 : ρ(A
1 )] = [A(I ) : ρ(A(I ))] = d(ρ)2 is finite by assumption, γ contains [34] the identity morphism, to wit there is V ∈ A
1 such that V x = ρρ(x)V for all x ∈ A
1 . Since ρρ is (e-)localized in I , Lemma 2.13 implies V ∈ A(I ), thus the equation V x = ρρ(x)V also holds for x ∈ A(I ), and strong additivity together with local normality of ρ, ρ imply that it holds for all x ∈ A∞ . Thus 1 = idA∞ ≺ ρρ, and ρ is a conjugate, in the sense of [36], of ρ in the tensor ∗-category G − Locf A. Choosing a conjugate or dual ρ for every ρ ∈ G − Locf A and duality morphisms e : ρ ⊗ ρ → 1, 1 → ρ ⊗ ρ satisfying the triangular equations we may consider G−Locf A as a spherical category. Remark 2.20. Every object ρ in a spherical or C ∗ -category with simple unit has a dimension d(ρ) living in the ground field, C in the present situation. This dimension of an object localized in I is related to the index by the following result of Longo [34]: d(ρ) = [A(I ) : ρ(A(I ))]1/2 .
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Summarizing the preceding discussion we have Theorem 2.21. G−Loc A is a braided crossed G-category and G−Locf A is a rigid semisimple braided crossed G-category. Remark 2.22. 1. It is obvious that for any braided G-crossed category D, the degree zero subcategory De is a braided tensor category. In the case at hand, Loc A = (G−Loc A)e is the familiar category of transportable localized morphisms defined in [20]. But for non-trivial symmetries G, the category G−Loc A contains information that cannot be obtained from Loc A. 2. The closest precedent to our above considerations can be found in [54]. There, however, several restrictive assumptions were made, in particular only abelian groups G were considered. Under these assumptions the G-crossed structure essentially trivializes. 2.4. Chiral conformal QFT on S 1 . In this subsection we briefly recall the main facts pertinent to chiral conformal field theories on S 1 and their representations, focusing in particular on the completely rational models introduced and analyzed in [29]. While nothing in this subsection is new, we include the material since it will be essential in what follows. Let I be the set of intervals in S 1 , i.e. connected open non-empty and non-dense subsets of S 1 . (I can be identified with the set {(x, y) ∈ S 1 × S 1 | x = y}.) For every J ⊂ S 1 , J is the interior of the complement of J . This clearly defines an involution on I. Definition 2.23. A chiral conformal field theory is a quadruple (H0 , A, U, ), usually simply denoted by A, where 1. H0 is a separable Hilbert space with a distinguished non-zero vector . 2. A is an assignment I I → A(I ), where A(I ) is a von Neumann algebra on H0 . 3. U is a strongly continuous unitary representation of the M¨obius group P SU (1, 1) = SU (1, 1)/{1, −1}, i.e. the group of those fractional linear maps C → C which map the circle into itself, on H0 . These data must satisfy • Isotony: I ⊂ J ⇒ A(I ) ⊂ A(J ), • Locality: I ⊂ J ⇒ A(I ) ⊂ A(J ) , • Irreducibility: ∨I ∈I A(I ) = B(H0 ) (equivalently, ∩I ∈I A(I ) = C1), • Covariance: U (a)A(I )U (a)∗ = A(aI ) ∀a ∈ P SU (1, 1), I ∈ I, • Positive energy: L0 ≥ 0, where L0 is the generator of the rotation subgroup of P SU (1, 1), • Vacuum: every vector in H0 which is invariant under the action of P SU (1, 1) is a multiple of . 2.24. For consequences of these axioms see, e.g., [22]. We limit ourselves to listing some facts: 1. Type: The von Neumann algebra A(I ) is a factor of type III (in fact III1 ) for every I ∈ I. 2. Haag duality: A(I ) = A(I ) ∀I ∈ I. 3. Reeh-Schlieder property: A(I ) = A(I ) = H0 ∀I ∈ I.
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4. The modular groups and conjugations associated with (A(I ), ) have a geometric meaning, cf. [6, 22] for details. 5. Additivity: If I, J ∈ I are such that I ∩ J, I ∪ J ∈ I then A(I ) ∨ A(J ) = A(I ∪ J ). In order to obtain stronger results we introduce two further axioms. Definition 2.25. Two intervals I, J ∈ I are called adjacent if their closures intersect in exactly one point. A chiral CFT satisfies strong additivity if I, J adjacent
0
⇒
A(I ) ∨ A(J ) = A(I ∪ J ).
A chiral CFT satisfies the split property if the map m : A(I ) ⊗alg A(J ) → A(I ) ∨ A(J ),
x ⊗ y → xy
extends to an isomorphism of von Neumann algebras whenever I, J ∈ I satisfy I ∩ J = ∅. Remark 2.26. By M¨obius covariance strong additivity holds in general if it holds for one pair I, J of adjacent intervals. Furthermore, every CFT can be extended canonically to one satisfying strong additivity. The split property is implied by the property T re−βL0 < ∞ ∀β > 0. The latter property and strong additivity have been verified in all known rational models. Definition 2.27. A representation π of A on a Hilbert space H is a family {πI , I ∈ I}, where πI is a unital ∗-representation of A(I ) on H such that I ⊂J
⇒
πJ A(I ) = πI .
(2.6)
π is called covariant if there is a positive energy representation Uπ of the universal covering group P SU (1, 1) of the M¨obius group on H such that Uπ (a)πI (x)Uπ (a)∗ = πaI (U (a)xU (a)∗ ) ∀a ∈ P SU (1, 1), I ∈ I. We denote by Rep A the C ∗ -category of all representations on separable Hilbert spaces, with bounded intertwiners as morphisms. Definition/Proposition 2.28. If A satisfies strong additivity and π is a representation then the Jones index of the inclusion πI (A(I )) ⊂ πI (A(I )) does not depend on I ∈ I and we define the dimension d(π) = [πI (A(I )) : πI (A(I ))]1/2 ∈ [1, ∞]. We define Repf A to be the full subcategory of Rep A of those representations satisfying d(π ) < ∞. As just defined, Rep A and Repf A are just C ∗ -categories. In order to obtain the well known result [20, 22] that the category of all (separable) representations can be equipped with braided monoidal structure, we need the following: Proposition 2.29. Every chiral CFT (H0 , A, U, ) satisfying strong additivity gives rise to a QFT on R.
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Proof. We arbitrarily pick a point ∞ ∈ S 1 and consider I∞ = {I ∈ I | ∞ ∈ I }. Identifying S 1 − {∞} with R by stereographic projection ∞ '$
&% we have a bijection between I∞ and K. The family A(I ), I ∈ K is just the restriction of A(I ), I ∈ I to I ∈ I∞ ≡ K. By 2.24, A satisfies Haag duality on S 1 , and together with strong additivity (on S 1 ) this implies Haag duality (on R) and strong additivity in the sense of Definition 2.1. Remark 2.30. The definition of G-actions on a chiral CFT on S 1 is analogous to Definition 2.3, Condition 1 now being required for all I ∈ I. Conditions 1-2 imply V (g)U (a) = U (a)V (g) ∀g ∈ G, a ∈ P SU (1, 1). (To see this observe that 1-2 imply that V (g) commutes with the modular groups associated with the pairs (A(I ), ) for any I ∈ I. By 2.24.4 the latter are one-parameter subgroups of U (P SU (1, 1)) which generate U (P SU (1, 1)).) Condition 3 now is equivalent to the more convenient axiom 3’. If U (g) ∈ C1 then g = e. (Proof. If U (g) ∈ C1 then αg = id, thus g = e by 3. Conversely, if αg acts trivially on some A(I ) then U (g) commutes with A(I ) and in fact with all A(I ) by V (g)U (a) = U (a)V (g). Thus the irreducibility axiom implies U (g) ∈ C1.) Given a CFT on S 1 and ignoring a possibly present G-action we have the categories Rep A (Repf A) as well as the braided tensor categories Loc A (Locf A) associated with the restriction of A to R. The following result, cf. [29, Appendix], connects these categories. Theorem 2.31. Let (H0 , A, U, ) be a chiral CFT satisfying strong additivity. Then there are equivalences of ∗-categories Loc A Rep A, Locf A Repf A, where Rep(f ) A refers to the chiral CFT and Definition 2.27, whereas Loc(f ) A refers to the QFT on R obtained by restriction and Definition 2.8. Proof. The strategy is to construct a functor Q : Loc A → Rep A of ∗-categories and to prove that it is fully faithful and essentially surjective. Let ρ ∈ Loc A be localized in I ∈ K ≡ I∞ . Our aim is to define a representation π = (πI , I ∈ I) on the Hilbert space H0 . For every J ∈ I∞ we define πJ = ρ A(J ), considered as a representation on H0 . If ∞ ∈ J we pick an interval K ∈ I∞ , K ∩ J = ∅. By transportability of ρ there exists ρ localized in K and a unitary u ∈ Hom(ρ, ρ ). Defining πJ = u∗ · u we need to show that πJ is independent of the choices involved. Thus let ρ
be localized in K (this may be assumed by making K large enough) and v ∈ Hom(ρ, ρ
), giving rise
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to πJ = v ∗ · v. Now, u ◦ v ∗ ∈ Hom(ρ
, ρ ), thus uv ∗ ∈ A(K) by Lemma 2.13, and therefore πJ (x) = v ∗ xv = v ∗ (vu∗ uv ∗ )xv = v ∗ vu∗ xuv ∗ v = u∗ xu = πJ (x), since x ∈ A(J ) ⊂ A(K) . Having defined πJ for all J ∈ I we need to show (2.6) for all I, J ∈ I. There are three cases of inclusions I ⊂ J to be considered: (i) I, J ∈ I∞ , (ii) I ∈ I∞ , J ∈ I∞ , (iii) I, J ∈ I∞ . Case (i) is trivial since πI = πJ = ρ, restricted to A(I ), A(J ) respectively. Case (iii) is treated by using K ⊂ J for the definition of both πI , πJ and appealing to the uniqueness of the latter. In case (ii) we have πJ = u∗ · u with u ∈ Hom(ρ, ρ ), ρ localized in K ⊂ J . For x ∈ A(I ) we have πJ (x) = u∗ xu = u∗ ρ (x)u = ρ(x) = πI (x), as desired. This completes the proof of π = {πJ } ∈ Rep A. Let ρ1 , ρ2 ∈ Loc A and let π1 , π2 be the corresponding representations. We claim that s ∈ Hom(ρ1 , ρ2 ) implies s ∈ Hom(π1 , π2 ). Let ∞ ∈ J , K ∈ I∞ , K ∩ J = ∅, ρi localized in K and ui ∈ Hom(ρi , ρi ) unitaries, such that then πJ,i = u∗i · ui . We have u2 su∗1 ∈ Hom(ρ1 , ρ2 ). Since both ρ1 , ρ2 are localized in K we have u2 su∗1 ∈ A(K) ⊂ A(J ) . Now the computation sπJ,1 (x) = su∗1 xu1 = u∗2 (u2 su∗1 )xu1 = u∗2 x(u2 su∗1 )u1 = u∗2 xu2 s = πJ,2 s shows s ∈ Hom(πJ,1 , πJ,2 ). Since this works for all J such that ∞ ∈ J we have s ∈ Hom(π1 , π2 ), and we have defined a faithful functor Q : Loc A → Rep A. Obviously, Q is faithful. In view of ρ = π A∞ it is clear that s ∈ Hom(π, π ) implies s ∈ Hom(ρ, ρ ), thus Q is full. Let now π ∈ Rep A and I ∈ I. Then πI is a unital ∗-representation of A(I ) on a separable Hilbert space. Since A(I ) is of type III and H0 is separable, πI is unitarily implemented. I.e. there exists a unitary u : H0 → Hπ such that πI (x) = uxu∗ for all x ∈ A(I ). Then (πJ ) = (u∗ πJ (·)u) is a representation on H0 that satisfies π ∼ = π and πI = πI,0 = id. Haag duality (on S 1 ) implies πJ (A(J )) ⊂ A(J ) whenever J ⊃ I . If we choose I such that ∞ ∈ I then πJ , J ⊃ I defines an endomorphism ρ of A∞ whose extension to a representation Q(ρ) coincides with π . Thus Q is essentially surjective and therefore an equivalence Loc A Rep A. Now, ρ ∈ Loc A is in Locf A iff d(ρ) = [A(I ) : ρ(A(I ))]1/2 < ∞ whenever ρ is localized in I . On the other hand, π ∈ Rep A is in Repf A iff d(π ) = [πI (A(I )) : πI (A(I ))]1/2 < ∞. In view of the above construction it is clear that d(π ) = d(ρ) if π is the representation corresponding to ρ. Thus Q restricts to an equivalence Locf A Repf A. Using the equivalence Q the braided monoidal structure of Loc(f ) A can be transported to Rep(f ) A: Corollary 2.32. Rep A (Repf A) can be equipped with a (rigid) braided monoidal structure such that there are equivalences Loc A Rep A, Locf A Repf A of braided monoidal categories.
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Remark 2.33. 1. It is quite obvious that the braided tensor structure on Rep A provided by the above constructions is independent, up to equivalence, of the choice of the point ∞ ∈ S 1 . For an approach to the representation theory of QFTs on S 1 that does not rely on cutting the circle see [21]. The latter, however, seems less suited for the analysis of G−Loc A for non-trivial G since the g-localized endomorphisms of A∞ do not extend to endomorphisms of the global algebra Auniv of [21] if g = e. 2. Given a chiral CFT A, the category Rep A is a very natural object to consider. Thus the significance of the degree zero category (G−Loc A)e is plainly evident: It enables us to endow Rep A with a braided monoidal structure in a considerably easier way than any known alternative. 3. By contrast, the rest of the category G−Loc A has no immediate physical interpretation. After all, the objects of (G − Locf A)g with g = e do not represent proper representations of A since they ‘behave discontinuously at ∞’. In fact, it is not difficult to prove that, given two adjacent intervals I, J ∈ I and g = e, there exists no representation π of A such that π A(I ) = id and π A(J ) = βg . Thus ρ, considered as a representation of A∞ , cannot be extended to a representation of A. The main physical relevance of G−Locf A is that – in contradistinction to Repf A – it contains sufficient information to compute Repf AG . This will be discussed in the next section. 4. On the purely mathematical side, the category G−Loc A may be used to define an invariant of three dimensional G-manifolds [60], i.e. 3-manifolds equipped with a principal G-bundle. As mentioned in the introduction, this provides an equivariant version of the construction of a 3-manifold invariant from a rational CFT. As is well known, there are models, like the U (1) current algebra, that satisfy the standard axioms including strong additivity and the split property and that have infinitely many inequivalent irreducible representations. Since in this work we are mainly interested in rational CFTs we need another axiom to single out the latter. Definition/Proposition 2.34. [29]. Let A satisfy strong additivity and the split property. Let I, J ∈ I satisfy I ∩ J = ∅ and write E = I ∪ J . Then the index of the inclusion A(E) ⊂ A(E ) does not depend on I, J and we define µ(A) = [A(E ) : A(E)] ∈ [1, ∞]. A chiral CFT on S 1 is completely rational if it satisfies (a) strong additivity, (b) the split property and (c) µ(A) < ∞. Remark 2.35. 1. Thus every CFT satisfying strong additivity and the split property comes along with a numerical invariant µ(A) ∈ [1, ∞]. The models where the latter is finite – the completely rational ones – are among the best behaved (non-trivial) quantum field theories, in that very strong results on both their structure and representation theory have been proven in [29]. In particular the invariant µ(A) has a nice interpretation. 2. All known classes of rational CFTs are completely rational in the above sense. For the WZW models connected to loop groups this is proven in [61, 63]. More importantly, the class of completely rational models is stable under tensor products and finite extensions and subtheories, cf. Sect. 3 for more details. This has applications to orbifold and coset models. Theorem 2.36. [29]. Let A be a completely rational CFT. Then • Every representation of A on a separable Hilbert space is completely reducible, i.e. a direct sum of irreducible representations. (For non-separable representations this
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is also true if one assumes local normality, which is automatic in the separable case, or equivalently covariance.) • Every irreducible separable representation has finite dimension d(π ), thus Repf A is just the category of finite direct sums of irreducible representations. • The number of unitary equivalence classes of separable irreducible representations is finite and dim Repf A = µ(A), where dim Repf A is the sum of the squared dimensions of the simple objects. • The braiding of Locf A Repf A is non-degenerate, thus Repf A is a unitary modular category in the sense of Turaev [59]. 3. Orbifold Theories and Galois Extensions 3.1. The restriction functor R : (G − LocA)G → LocAG . After the interlude of the preceding subsection we now return to QFTs defined on R with symmetry G. (Typically they will be obtained from chiral CFTs on S 1 by restriction, but in the first subsections this will not be assumed.) Our aim is to elucidate the relationship between the categories G−Loc A and Loc AG , where AG is the ‘orbifold’ subtheory of G-fixpoints in the theory A. Definition 3.1. Let (H, A, ) be a QFT on R with an action (in the sense of Definition 2.3) of a compact group G. Let H0G and A(I )G be the fixpoints under the G-action on H0 and A(I ), respectively. Then the orbifold theory AG is the triple (H0G , AG , ), where AG (I ) = A(I )G H0G . Remark 3.2. 1. The definition relies on ∈ H0G and A(I )G H0G ⊂ H0G for all I ∈ K. Denoting by p the projector onto H0G , we have AG (I ) = A(I )G H0G = pA(I )p, where the right hand side is understood as an algebra acting on pH0 = H0G . Furthermore, since A(I )G acts faithfully on H0G we have algebra isomorphisms A(I )G ∼ = AG (I ). G G 2. It is obvious that the triple (H0 , A , ) satisfies isotony and locality. Irreducibility follows by ∨I ∈K AG (I ) = p(∨I ∈K A(I ))p together with ∨I A(I ) = B(H0 ). However, strong additivity and Haag duality of the fixpoint theory are not automatic. For the time being we will postulate these properties to hold. Later on we will restrict to settings where this is automatically the case. 3.3. For later purposes we recall a well known fact about compact group actions on QFTs in the present setting. Namely, for every I ∈ K, the G-action on A(I ) has full of irreducible G-spectrum, [15]. This means that for every isomorphism class α ∈ G representations of G there exists a finite dimensional G-stable subspace Vα ⊂ A(I ) on which the G-action restricts to the irrep πα . Vα can be taken to be a space of isometries of support 1. (This means that Vα admits a basis {vαi , i = 1, . . . , dα } such that i i∗ i∗ j G i vα vα = 1 and vα vα = δij 1.) Furthermore, A(I ) is generated by A(I ) and the spaces Vα , α ∈ G. These observations have an important consequence for the representation categories of fixpoint theories [15]. Namely the category Locf AG contains a full symmetric subcategory S equivalent to the category Repf G of finite dimensional continuous unitary representations of G. The objects in S are given by the localized endomorphisms of AG ∞
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∗ of the form ρα (·) = i vαi · vαi , where {vαi } is a space of isometries with support 1 in (Equivalently, a simple object of ρ ∈ Locf AG A(I ) transforming under the irrep α ∈ G. is in S iff the corresponding representation π0 ◦ ρ of AG is contained in the restriction to AG of the defining (or vacuum) representation of A.) 3.4. We now begin our study of the relationship between G − LocA and LocAG . Let (G−LocA)G denote the G-invariant objects and morphisms of G−LocA. By definition of the G-action on G−LocA, ρ ∈ (G−LocA)G implies ρ ◦ βg = βg ◦ ρ for all g ∈ G, G thus ρ(AG ∞ ) ⊂ A∞ . Every ρ ∈ G−LocA is G-localized in some interval I . In view of Definition 2.8 it is obvious that the restriction ρ AG ∞ acts trivially on A(J ) not only if J < I , but also if J > I . Thus ρ AG is a localized endomorphism of AG ∞ ∞. G Furthermore, if ρ, σ ∈ (G − LocA) and s ∈ Hom(G−LocA)G (ρ, σ ) it is easy to see G G G that s ∈ HomLocAG (ρ AG ∞ , σ A∞ ). This suggests that ρ A∞ ∈ LocA . HowG ever, this also requires showing that the restricted morphism ρ A∞ is transportable by morphisms in LocAG . This requires some work. G Proposition 3.5. Let ρ ∈ (G−LocA)G . Then ρ AG ∞ ∈ LocA .
Proof. By definition, ρ is G-localized in some interval I . As we have seen in 3.4, ρ AG ∞ is localized in I , and it remains to show that ρ AG ∞ is transportable. Let thus J be another interval. By transportability of ρ ∈ G−LocA, there exists ρ that is G-localized in J and a unitary u ∈ HomG−LocA (ρ, ρ ). Define ρ g = γg ( ρ ) = βg ◦ ρ ◦ βg−1 . Since γg is an automorphism of G−LocA and ρ is G-invariant we have γg (u) := βg (u) ∈ HomG−LocA (ρ, ρ g ). Defining vg = βg (u)u∗ we have vgh = βgh (u)u∗ = βg (vh )βg (u)u∗ = βg (vh )vg
∀g, h.
ρ, ρ g ), and since all ρ g are G-localized in J , Lemma 2.13 Furthermore, vg ∈ Hom( implies vg ∈ A(J ). Thus g → vg is a (strongly continuous) 1-cocycle in A(J ). Since A(J ) is a type III factor and the G-action has full G-spectrum, there exists [57] a unitary w ∈ A(J ) such that vg = βg (w)w ∗ for all g ∈ G. Defining ρ = Ad w∗ ◦ ρ , we have w∗ u ∈ Hom(ρ, ρ ). Now, βg (u)u∗ = βg (w)w ∗ is equivalent to βg (w ∗ u) = w ∗ u, thus w∗ u is G-invariant. Together with the obvious fact that ρ is G-localized in J , this implies G ρ AG ∞ ∈ LocA . G Corollary 3.6. Restriction to AG ∞ provides a strict tensor functor R : (G−LocA) → G LocA which is faithful on objects and morphisms.
Proof. With the exception of faithfulness, which follows from the isomorphisms A(I )G ∼ = AG (I ), this is just a restatement of our previous results. Remark 3.7. 1. In Subsect. 3.4 we will show that R, when restricted to (G−Locf A)G , is also surjective on morphisms (thus full) and objects. Thus R will establish an isomorphism (G−Locf A)G ∼ = Locf AG . 2. We comment on our definition of the fixpoint category C G of a category C under a Gaction. In the literature, cf. [58, 30, 31], one can find a different notion of fixpoint category, which we denote by CG for the present purposes. Its objects are pairs (X, {ug , g ∈ G}), where X is an object of C and the ug ∈ HomC (X, γg (X)) are isomorphisms making the left diagram in Fig. 1 commute. The morphisms between (X, {ug , g ∈ G}) and (Y, {vg , g ∈ G}) are those s ∈ HomC (X, Y ) for which the right diagram in Fig. 1
Conformal Orbifold Theories and Braided Crossed G-Categories
X
ug
- γg (X)
@ @ ugh @ R @
γg (uh )
?
γgh (X)
X
745 s Y
ug
vg
? γg (X)
? - γg (Y )
γg (s)
Fig. 1. Objects and Morphisms of CG
commutes. (According to J. Bernstein, CG should rather be called the category of Gmodules in C.) It is clear that C G can be identified with a full subcategory of CG via X → (X, {id}), but in general this inclusion need not be an equivalence. However, it is an equivalence in the case of C = G−LocA. To see this, let (ρ, {ug }) ∈ (G−LocA)G . Assume ρ is G-localized in I . By definition of (G−LocA)G , g → ug is a 1-cocycle in A(I ), and by the above discussion there exists w ∈ A(I ) such that ug = βg (w)w ∗ for all g ∈ G. Defining ρ = Ad w∗ ◦ ρ, an easy computation shows ρ ∈ (G−LocA)G . G Since w : ρ → ρ is an isomorphism, the inclusion (G−LocA) → (G−LocA)G is essentially surjective, thus an equivalence. 3.2. The extension functor E : LocAG → (G−LocA)G . In view of Remark 3.2 we are in a setting where both A = (H0 , A(·), ) and AG = (H0G , AG (·), ) are QFTs on R. In this situation it is well known that there exists a monoidal functor E : Loc AG → End A∞ from the tensor category of localized transportable endomorphisms of the subtheory AG to the (not a priori localized) endomorphisms of the algebra A∞ . There are essentially three ways to construct such a functor. First, Roberts’ method of localized cocycles, cf. e.g. [55], which is applicable under the weakest set of assumptions. (Neither finiteness of the extension nor factoriality or Haag duality are required.) Unfortunately, in this approach it is relatively difficult to make concrete computations, cf. however [8]. Secondly, the subfactor approach of Longo and Rehren [35] as further studied by Xu, B¨ockenhauer and Evans, cf. e.g. [62, 4]. This approach requires factoriality of the local algebras and finiteness of the extension, but otherwise is very powerful. Thirdly, there is the approach of [42], which assumes neither factoriality nor finiteness, but which is restricted to extensions of the form AG ⊂ A. For the present purposes, this is of course no problem. Theorem 3.8. [42]. Let A = (H0 , A(·), ) be a QFT on R with G-action such that AG = (H0G , AG (·), ) is a QFT on R. There is a functor E : Loc AG → End A∞ with the following properties: 1. For every ρ ∈ Loc AG we have that E(ρ) commutes with the G-action β, i.e. E(ρ) ∈ (End A∞ )G . The restriction E(ρ) AG ∞ coincides with ρ. On the arrows, E is the inclusion AG ∞ → A∞ . Thus E is faithful and injective on the objects. 2. E is strict monoidal. (Recall that Loc AG and End A∞ are strict.) 3. If ρ is localized in the interval I ∈ K then E(ρ) is localized in the half-line (inf I, +∞). This requirement makes E(ρ) unique. Remarks on the proof. Fix an interval I ∈ K. By 3.3, we can find a family {Vα ⊂ of finite dimensional subspaces of isometries of support 1 on which the A(I ), α ∈ G}
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Now the algebra A(I ) is genG-action restricts to the irreducible representation α ∈ G. and A∞ is generated by AG erated by A(I )G and the family {Vα , α ∈ G}, ∞ and the family Furthermore, σα = i vαi · vαi ∗ is a transportable endomorphism of AG {Vα , α ∈ G}. ∞ localized in I , thus σα ∈ Locf AG . Now E(ρ) is determined by Rehren’s prescription [53]: ρ(x) x ∈ AG ∞ E(ρ)(x) = c(σα , ρ)x x ∈ Vα , where c(σα , ρ) is the braiding of the category Loc AG . (The proof of existence and uniqueness of E(ρ) is given in [42], generalizing the automorphism case treated in [17]. Note that despite the appearances this definition of E does not depend on the chosen spaces Vα .) On the arrows HomLoc AG (ρ, σ ) ⊂ AG ∞ we define E via the inclusion AG → A . For the verification of all claimed properties see [42, Prop. 3.11]. ∞ ∞ Remark 3.9. 1. The definition of E does not require d(ρ) < ∞. But from now on we will restrict E to the full subcategory Locf AG ⊂ Locf AG . 2. The extension functor E is faithful but not full. Our aim will be to compute HomEnd A∞ (E(ρ), E(σ )), but this will require some categorical preparations. 3.3. Recollections on Galois extensions of braided tensor categories. From the discussion in 3.3 it is clear that the extension E(ρ) ∈ End A∞ is trivial, i.e. isomorphic to a direct sum of dim(ρ) ∈ N copies of the tensor unit 1, for every ρ in the full symmetric subcategory S. It is therefore natural to ask for the universal faithful tensor functor ι : C → D that trivializes a full symmetric subcategory S of a rigid braided tensor category C. Such a functor has been constructed independently in [44] (without explicit discussion of the universal property) and in [5]. (The motivation of both works was to construct a modular category from a non-modular braided category by getting rid of the central/degenerate/transparent objects.) A universal functor ι : C → D trivializing S exists provided every object in S has trivial twist θ (X), both approaches relying on the fact [18, 9] that under this condition S is equivalent to the representation category of a group G, which is finite if S is finite and otherwise compact [18] or proalgebraic [9]. In the subsequent discussion we will use the approach of [44] since it was set up with the present application in mind, but we will phrase it in the more conceptual way expounded in [48]. Given a rigid symmetric tensor ∗-category S with simple unit and trivial twists, the main result of [18] tells us that there is a compact group G such that S Repf G. (In our application to the subcategory S ⊂ Locf AG for an orbifold CFT AG we don’t need to appeal to the reconstruction theorem since the equivalence S Repf G is proven already in [15].) Assuming S (and thus G) to be finite we know that there is a commutative strongly separable Frobenius algebra (γ , m, η, , ε) in S, where γ corresponds to the left regular representation of G under the equivalence. See [46] for the precise definition and proofs. (More generally, this holds for any finite dimensional semisimple and cosemisimple Hopf algebra H [46]. For infinite compact groups and infinite dimensional discrete quantum groups one still has an algebra structure (γ , m, η), cf. [50].) The group G can be recovered from the monoid structure (γ , m, η) as G∼ = {s ∈ Endγ | s ◦ m = m ◦ s ⊗ s, s ◦ η = η}.
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Now we define [48] a category C 0 S with the same objects and same tensor product of objects as C, but larger hom-sets: HomC0 S (ρ, σ ) = HomC (γ ⊗ ρ, σ ). The compositions ◦, ⊗ of morphisms are defined using the Frobenius algebra structure on γ . Finally, C S is defined as the idempotent completion (or Karoubian envelope) of C 0 S. The latter contains C 0 S as a full subcategory and is unique up to equivalence, but there also is a well known canonical model for it. I.e., the objects of C S are pairs (ρ, p), where ρ ∈ C 0 S and p = p2 = p∗ ∈ EndC0 S (ρ). The morphisms are given by HomCS ((ρ, p), (σ, q)) = q ◦ HomC0 S (ρ, σ ) ◦ q = {s ∈ HomC0 S (ρ, σ ) | s = q ◦ s ◦ p}. The inclusion functor ι : C → C S, ρ → (ρ, idρ ) has the desired trivialization property since dim HomCS (1, ι(ρ)) = d(ρ) for all ρ ∈ S. The group G acts on a morphism s ∈ HomCS ((ρ, p), (σ, q)) ⊂ HomC (γ ⊗ ρ, σ ) via γg (s) = s ◦ g −1 ⊗ idρ , where g ∈ Aut(γ , m, η) ∼ = G. The G-fixed subcategory (C S)G is just the idempotent completion of C and thus equivalent to C. The braiding c of C lifts to a braiding of C S iff all objects of S are central, i.e. c(ρ, σ )c(σ, ρ) = id for all ρ ∈ S and σ ∈ C. This, however, will not be the case in the application to QFT. As shown in [48], in the general case C S is a braided crossed G-category. We need one concrete formula from [48]. Namely, if p ∈ EndCS (ρ) ∼ = HomC (γ ⊗ ρ, ρ) is such that (ρ, p) ∈ C S is simple, then the morphism γ ∂(ρ, p) =
p
e η
−1 ·
@
@
@
@
ρ
@
(3.1) @ p
ρ γ
is an automorphism of the monoid (γ , m, η), thus an element of G. We note for later use that the numerical factor (· · · )−1 is d(ρ, p)−1 and that replacing the braidings by their @ @ duals ( @ ↔ ) gives the inverse group element. @ @ If the category S, equivalently the group G are infinite, the above definition of C S needs to be reconsidered since, e.g., the proof of semisimplicity must be modified. The original construction of C S in [44] does just that. Using the decomposition γ ∼ = ⊕i∈G d(γi )γi of the regular representation one defines HomC (γi ⊗ ρ, σ ) ⊗ Hi , (3.2) HomC0 S (ρ, σ ) = i∈G
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where F : S → Repf G is an equivalence, γi ∈ S is such that F (γi ) ∼ = πi and Hi is the representation space of the irreducible representation πi of G. (It is easily seen that HomC0 S (ρ, σ ) is finite dimensional for all ρ, σ ∈ C.) Now the compositions ◦, ⊗ of morphisms are defined by the formulae m
s ψk ◦ t ψl =
Nkl
mα mα ∗ s ◦ idγk ⊗ t ◦ wkl ⊗ idρ K(wkl ) (ψk ⊗ ψl ),
α=1 m∈G m
u ψk ⊗ w ψl =
Nkl
mα u ⊗ v ◦ idγk ⊗ ε(γl , ρ1 ) ⊗ idρ2 ◦ wkl
α=1 m∈G mα ∗ ) (ψk ⊗ ψl ), ⊗idρ1 ρ2 K(wkl
ψk ∈ Hk , ψl ∈ Hl , t ∈ Hom(γl ⊗ ρ, σ ), s ∈ Hom(γk ⊗ σ, δ) and where k, l ∈ G, u ∈ Hom(γk ⊗ ρ1 , σ1 ), V ∈ Hom(γl ⊗ ρ2 , σ2 ). For for further details and the definition of the ∗-involution, which we don’t need here, we refer to [44]. For finite G it is readily verified that the two definitions of C S given above produce isomorphic categories. If is central in C, equivalently c(ρ, σ )c(σ, ρ) = id for all ρ ∈ S, σ ∈ C, then C S inherits the braiding of C, cf. [44]. If this is not the case, − Mod is only a braided crossed G-category [48]. Before we return to our quantum field theoretic considerations we briefly comment on the approach of [5] and the related works [52, 32, 30, 31]. As before, one starts from the (Frobenius) algebra in S corresponding to the left regular representation of G. One now considers the category − Mod of left modules over this algebra. As already observed in [52], this is a tensor category. Again, if is central in C then − Mod is braided [5], whereas in general − Mod is a braided crossed G-category [30, 31]. (The braided degree zero subcategory coincides with the dyslexic modules of [52].) In [48] an equivalence of C S and − Mod is proven. In the present investigations it is more convenient to work with C S since it is strict if C is. 3.4. The isomorphism Locf AG ∼ = (G−Locf A)G . In Subsect. 3.2, the extension functor E was defined on the entire category Loc AG . It is faithful but not full, and our aim is to obtain a better understanding of HomA∞ (E(ρ), E(σ )). From now on we will restrict it to the full subcategory Locf AG of finite dimensional (thus rigid) objects, and we abbreviate C = Locf AG throughout. Furthermore, S ⊂ C will denote the full subcategory discussed in 3.3. We recall that S Repf G as symmetric tensor category. Since the definition of C S in [44] was motivated by the formulae [53, 42] for the intertwiner spaces HomA∞ (E(ρ), E(σ )), the following is essentially obvious: Proposition 3.10. Under the same assumptions on A and AG and notation as above, the functor E : C → (End A∞ )G factors through the canonical inclusion functor ι : C → C S, i.e. there is a tensor functor F : C S → End A∞ such that C
ιCS @ @ F E@ @ ? R End A∞
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commutes. (Note that F (C S) ⊂ (End A∞ )G .) The functors E : C → (End A∞ )G , F : C S → End A∞ are faithful and full. Proof. First, we define F on the tensor category C 0 S of [44, 48], which has the same objects as C but larger hom-sets. We clearly have to put F (ρ) := E(ρ). Now fix an interval I ∈ K and subspaces Hi ⊂ A(I ) of isometries on which G acts according to the irrep πi . Let γi be the endomorphism of AG ∞ implemented by Hi . As stated in [53] and proven in [42], the intertwiner spaces between extensions E(ρ), E(σ ) is given by HomA∞ (E(ρ), E(σ )) = spani∈G Hom C (γi ρ, σ )Hi ⊂ A∞ . On the one hand, this shows that every G-invariant morphism s ∈ HomG−Locf A (E(ρ), E(σ )) is in HomLocf AG (ρ, σ ), implying that E : C → (End A∞ )G is full. On the other hand, it is clear that these spaces can be identified with those in the second definition (3.2) of HomC0 S (ρ, σ ). Under this identification, the compositions ◦, ⊗ of morphisms in C 0 S go into those in the category End A∞ as given in Definition 2.5, as is readily verified. Thus we have a full and faithful strict tensor functor F0 : C 0 S → End A∞ such that F0 ◦ ι = E. Now, C S is defined as the completion of C 0 S with splitting idempotents. Since the category End A∞ has splitting idempotents, the functor F0 extends to a tensor functor F : C S → End A∞ , uniquely up to natural isomorphism of functors. However, we give a more concrete prescription. Let (ρ, p) be an object of C S, i.e. ρ ∈ Locf A and p = p2 = p ∗ ∈ EndC0 S (ρ). Let I ⊂ K be an interval in which ρ ∈ Locf A is localized. Then Haag duality implies p ∈ A(I ). Since A(I ) is a type III factor (with separable predual) we can pick v ∈ A(I ) such that vv ∗ = p and v ∗ v = 1. Now we define F ((ρ, p))(·) = v ∗ F (ρ)(·)v ∈ End A∞ . This is an algebra endomorphism of A∞ since vv ∗ = p ∈ HomA∞ (E(ρ), E(ρ)). With this definition, the functor F : C S → End A∞ is strongly (but not strictly) monoidal. In [48] it was shown that C S is a braided crossed G-category. In view of the results of Sect. 2 it is natural to expect that the functor F actually takes its image in G−LocA and is a functor of braided crossed G-categories. In fact: Proposition 3.11. Let A = (H0 , A(·), ) be as before and G finite. Then (i) for every ρ ∈ Locf AG we have E(ρ) ∈ G−Locf A, thus the extension E(ρ) is a finite direct sum of endomorphisms ηi of A∞ that act as symmetries βgi on a half line [a, +∞). (ii) F (C S) ⊂ G−Locf A and F : C S → G−Locf A is a functor of G-graded categories, i.e. F ((C S)g ) ⊂ (G−Locf A)g for all g ∈ G. Proof. Claim (i) clearly follows from (ii). In order to prove the latter it is sufficient to show for every irreducible object (ρ, p) ∈ C S that E((ρ, p)) ∈ End A∞ is ∂(ρ, p)-localized. Let thus ρ ∈ C = Locf AG be localized in the interval I ∈ K and let p = p2 = p ∗ be a minimal projection in EndC0 S (ρ). Recall that F ((ρ, p)) is defined as v ∗ E(ρ)(·)v, where v ∈ A∞ satisfies vv ∗ = F (p), v ∗ v = 1. We may assume that v ∈ A(I ). Let J ∈ K such that I < J and let Hγ ⊂ A(J ) be a subspace of isometries transforming under the left regular representation of G. (I.e., we have isom ∗ = 1, v ∗ v = δ etries vg ∈ A(I ), g ∈ G such that βk (vg ) = vkg , v v g,h 1.) Let g h g v g
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γ (·) = g vg ·vg∗ ∈ End AG ∞ be the localized endomorphism implemented by Hγ . Thus Hγ = HomA (1, E(γ )). Now by Theorem 3.8 we have, for x ∈ Hγ ,
F ((ρ, p))(x) = v ∗ c(γ , ρ)xv = [E(γ )(v ∗ )c(ρ, γ )c(γ , ρ)E(γ )(v)] x,
where we have used (i) xv = vx (since x, v are localized in the disjoint intervals I, J , respectively), (ii) c(ρ, γ ) = 1 (follows by Lemma 2.17 since the localization region of ρ is in the left complement of the localization region of γ ) and (iii) E(γ )(v) = v (since v ∈ A(I ), on which E(ρ) acts trivially). This expression defines an element of Hom∗A (F ((ρ,∗p), F (γ )F ((ρ, p)). If v1 , . . . , v|G| ∈ HomA (1, E(γ )) are such that i vi vi = 1, vi vj = δi,j 1 then
vi∗ [E(γ )(v ∗ )c(ρ, γ )c(γ , ρ)E(γ )(v)] x ∈ EndA (F (ρ, p)).
By irreducibility of F ((ρ, p)) this expression is a multiple of idF (ρ,p) , thus
d((ρ, p)) F ((ρ, p))(x) = d(ρ, p) [E(γ )(v ∗ )c(ρ, γ )c(γ , ρ)E(γ )(v)] x = vi T r(ρ,p) (vi∗ [E(γ )(v ∗ )c(ρ, γ )c(γ , ρ)E(γ )(v)] x) i
γ
γ
v∗
@
@
@
@ =@
(ρ, p)
ρ @ v
γ x
@
ρ
ρ @
=
p
γ x
Conformal Orbifold Theories and Braided Crossed G-Categories
751
Now we express this as a diagram in C in terms of the representers x ∈ HomC (γ , γ ) and p ∈ HomC (γ ⊗ ρ, ρ). By definition of C S we obtain γ
γ
@
x
@
@
@
ρ
d((ρ, p)) F ((ρ, p))(x) =
@ p
A A
=
x
γ
@
@ ρ
@ p
γ
where we have used the commutativity = c(γ , γ ) ◦ . Thus by (3.1) and [48] we have F ((ρ, p))(x) = x ◦ ∂((ρ, p))−1 , where ∂((ρ, p)) ∈ Aut(γ , m, η) is the degree of (ρ, p). Recalling that the action of g ∈ Aut(γ , m, η) on the morphism s ∈ HomC (γ ⊗ρ, σ ) ∼ = HomCS (ρ, σ ) was defined as γg (s) = s ◦ g −1 ⊗ idρ , we see that F ((ρ, p))(x) = γ∂(ρ,p) (x). Thus F ((ρ, p)) ∈ End A∞ is ∂(ρ, p)-localized in the sense of Sect. 2, as claimed. Transportability of E((ρ, p)) follows from transportability of ρ. Thus E((ρ, p)) ∈ G − Locf A, and the same clearly follows for the non-simple objects of Locf A. The above computations have also shown that the functor F respects the G-gradings of C S and G−Loc A in the sense that F ((C S)g ) ⊂ (G−Locf A)g for all g ∈ G. The following result, which shows that Locf AG can be computed from G−Locf A, was the main motivation for this paper: Theorem 3.12. If G is finite then the functors E: Locf AG → (G−Locf A)G , G Locf AG R : (G−Locf A) → are mutually inverse and establish an isomorphism of strict braided tensor categories. Proof. By Subsect. 3.2, E : Locf AG → (EndA)G is a faithful strict tensor functor, which is full by Proposition 3.10. By Proposition 3.11 it takes its image in (G−Locf A)G . By Theorem 3.8 we have R ◦ E = idLocf AG , and E ◦ R = id(G−Locf A)G follows since ρ ∈ (G−Locf A)G is the unique right-localized extension to A∞ of R(ρ) = ρ AG ∞. Therefore E is surjective on objects and thus an isomorphism. That the braidings of Locf AG and (G−Locf A)G is clear in view of their construction.
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Remark 3.13. 1. The ‘size’ of Locf AG will be determined in Corollary 3.16. 2. Clearly the above is a somewhat abstract result, and in concrete models hard work is required to determine the category G−Locf A of twisted representations. (For a beautiful analysis of orbifolds of affine models in the present axiomatic setting see the series of papers [64, 37, 27].) However, Theorem 3.12 can be used to clarify the structure of Locf AG quite completely in the holomorphic case, cf. Subsect. 4.2. 3. Proposition 3.11 and Theorem 3.12 remain true when G is compact infinite. In order to see this one needs to show that C S is a braided crossed G-category also in the case of infinite S. In view of the fact that the existence of C S as a rigid tensor category with G-action was already established in [44] this can be done by an easy modification of the approach used in [48]. Then the proof of Proposition 3.10 easily adapts to arbitrary compact groups. 3.5. The equivalence Locf AG S G − Locf A. Our next aim is to show that the functor F gives rise to an equivalence Locf AG S G−Locf A of braided crossed G-categories. (Even though both categories are strict as monoidal categories and as Gcategories, the functor F will not be strict.) For the well known definition of a non-strict monoidal functor we refer, e.g., to [40]. Proposition 3.14. If G is finite then the functor F : C S → G−Locf A is essentially surjective, thus a monoidal equivalence. Proof. The bulk of the proof coincides with that of [42, Prop. 3.14], which remains essentially unchanged. We briefly recall the construction. Pick an interval I ∈ K. Since the G-action on A(I ) has full spectrum we can find isometries vg ∈ A(I ), g ∈ G, satisfying
vg vg∗ = 1,
vg∗ vh = δg,h 1,
βg (vh ) = vgh .
g
If now ρ ∈ G−Locf A is simple then it is easily verified that ρ (·) =
vg βg ρβg −1 (·) vg∗ ∈ G−Locf A
g
∈ (G−Locf A)G . Therefore ρ restricts to AG , and ρ AG commutes with all βg , thus ρ is localized in some interval, as was noted before. In order to show that ρ AG is transportable, let J be some interval, let σ be G-localized in J and let s : ρ → σ be unitary. Choosing isometries wg ∈ A(J ) as before and defining σ in analogy to ρ and ∗ G writing s˜ = g wg βg (s)vg , one easily verifies that s˜ is a unitary in Hom( ρ, σ ) . Thus G . As in [42] one now verifies ρ AG is transportable and defines an object of Loc A f ∞ that ρ = E( ρ AG ). Combined with the obvious fact ρ ≺ ρ this implies that every simple object ρ ∈ G−Locf A is a direct summand of E( ρ AG ) = F (ι( ρ AG )). In view of Proposition 3.10 and the fact that C S has splitting idempotents we conclude that ρ ∼ ρ AG ) ∈ C S. This implies that F is = F (σ ) for some subobject σ of ι( essentially surjective, thus an equivalence, which can be made monoidal, see e.g. [56].
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Remark 3.15. In Minkowski spacetimes of dimension ≥ 2 + 1, where there are no gtwisted representations, the functor E can be shown to be an equivalence under the weaker assumption that G is second countable, i.e. has countably many irreps, cf. [8]. Returning to the present one-dimensional situation, it is clear from the definition of E that E(Locf AG ∩S ) ⊂ Locf A = (G−Locf A)e , thus those ρ ∈ Locf AG which satisfy cρ,σ cσ,ρ = id for all σ ∈ S have a localized extension E(ρ). Its seems reasonable to expect that the restriction of F to the subcategory of C S generated by ι(C ∩ S ) is an equivalence with Locf A whenever G is second countable. We have refrained from going into this question since we are interested in the larger categories Locf AG and G−Locf A, and – in contradistinction to E : Locf AG → (G−Locf A)G – the functor F : Locf A S → G−Locf A is almost never essentially surjective (thus an equivalence). The point is that for ρ ∈ Locf AG we have E(ρ) ∼ = ⊕i ρi , where the ρi are gi -localized and the gi exhaust a whole conjugacy class since E(ρ) is G-invariant. Since the direct sum is finite, we see that the image of E : C S → G−Locf A can contain only objects σ whose degree ∂σ belongs to a finite conjugacy class. Since ‘most’ infinite non-abelian compact groups have infinite conjugacy classes, F will in general not be essentially surjective. (At least morally this is related to the fact [33] that the quantum double of a compact group G admits infinite dimensional irreducible representations whenever G has infinite conjugacy classes.) If, on the other hand, we consider E(ρ), where d(ρ) = ∞, the analysis of E(ρ) becomes considerably more complicated. Corollary 3.16. Under the assumptions of Theorem 3.18 we have dim Locf AG = |G| dim G−Locf A. Proof. Follows from G−Locf A ∼ = Locf AG S and dim C S = dim C/ dim S = dim C/|G|, cf. [44]. In order to prove the equivalence G − Locf A Locf AG S of braided crossed G-categories we need to consider the G-actions and the braidings. For the general definition of functors of G-categories we refer to [58], see also [7] and the references given there. Since our categories are strict as tensor categories and as G-categories, i.e. γgh (X) = γg ◦ γh (X) ∀g, h, X, γg (X ⊗ Y ) = γg (X) ⊗ γg (Y ) ∀g, X, Y,
(3.3)
we can simplify the definition accordingly: Definition 3.17. A functor F : C → C of categories with strict actions γg , γg of a group G is a functor together with a family of natural isomorphisms η(g) : F ◦ γg → γg ◦ F such that η(gh)X - γ ◦ F (X) F ◦ γgh (X) gh ||| ||| ||| ||| ||| ||| ||| ||| |||
F ◦ γg ◦ γh (X) γ ◦ F ◦ γh (X) γ ◦ γh ◦ F (X) η(g)γh (X) g γg (η(h)X ) g commutes. (There is no further condition on F if C, C , γ , γ are monoidal.) A functor of braided crossed G-categories is a monoidal functor of G-categories that respects the gradings and satisfies F (cX,Y ) = cF (X),F (Y ) for all X, Y ∈ C.
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Theorem 3.18. Let A = (H0 , A(·), ) be as before and G finite. Then F : C S → G−Locf A is an equivalence of braided crossed G-categories. Proof. It only remains to show that F is a functor of G-categories and that it preserves the braidings. Let (ρ, p) ∈ C S. Then βg ((ρ, p)) = (ρ, βg (p)), where βg (p) is the obvious G-action on C 0 S. Recall that F ((ρ, p)) ∈ End A∞ was defined as ∗ ∗ E(ρ)(·)v(ρ,p) , where v(ρ,p) ∈ A∞ satisfies v(ρ,p) v(ρ,p) = E(p). (For p = 1 v(ρ,p) we choose v(ρ,p) = 1.) Since E(ρ) commutes with γg we have γg (F ((ρ, p))) = γg (v(ρ,p) )∗ E(ρ)(·)γg (v(ρ,p) ). Because of γg (v(ρ,p) )γg (v(ρ,p) )∗ = γg (p), the isometries γg (v(ρ,p) ) and vβg (ρ,p) have the same range projection. Thus η(g)(ρ,p) = γg (v(ρ,p) )∗ v(ρ,βg (p)) is unitary and one easily verifies η(g)(ρ,p) ∈ Hom(F ◦βg (ρ, p), γg ◦ F (ρ, p)) as well as the commutativity of the above diagram. It remains to show that the functor F preserves the braidings. We first show that F (cρ,σ ) = cF (ρ),F (σ ) holds if ρ, σ ∈ C = Locf AG . By Theorem 3.8, E(ρ), E(σ ) are G-invariant, thus by the G-covariance of the braiding we have cE(ρ),E(σ ) ∈ AG ∞ . Thus the braiding of E(ρ), E(σ ) as constructed in Sect. 2 restricts to a braiding of ρ, σ and by uniqueness of the latter this restriction coincides with cρ,σ . Thus cE(ρ),E(σ ) = E(cρ,σ ) as claimed. The general result now is an obvious consequence of the naturality of the braidings of C S and of G−Locf A together with the fact that every object of C S and of G−Locf A is a subobject of one in C and (G−Locf A)G , respectively. 4. Orbifolds of Completely Rational Chiral CFTs 4.1. General theory. So far, we have considered an arbitrary QFT A on R subject to the technical condition that also AG be a QFT on R, some of the results assuming finiteness of G. The situation that we are really interested is the one where A derives from a chiral QFT on S 1 by restriction to R. Recall that in that case Loc(f ) AG has a ‘physical’ interpretation as a category Rep(f ) A of representations. Proposition 4.1. Let A be a completely rational chiral QFT with finite symmetry group G. Then the restrictions to R of A and AG are QFTs on R. Proof. In view of the discussion in Subsect. 2.4 it suffices to know that the chiral orbifold theory AG on S 1 satisfies strong additivity. In [64] it was proven that finite orbifolds of completely rational chiral QFTs are again completely rational, in particular strongly additive. Applying the results of [29] we obtain: Theorem 4.2. Let (H0 , A, ) be a completely rational chiral CFT and G a finite symmetry group. Then the braided crossed G-category G−Locf A has full G-spectrum, i.e. for every g ∈ G there is an object ρ ∈ G−Locf A such that ∂ρ = g. Furthermore, for every g ∈ G we have (dim ρ)2 = (dim ρ)2 = µ(A), ρ∈(G−Locf A)g
ρ∈Repf A
where the sums are over the equivalence classes of irreducible objects of degree g and e, respectively.
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Proof. By [64], the fixpoint theory AG is completely rational, thus by [29] the categories Repf AG ∼ = Locf AG are modular. Now, G−Locf A ∼ = Locf AG S, and fullness of the G-spectrum follows by [48, Cor. 3.27]. The statement on the dimensions follows from [48, Prop. 3.23]. Remark 4.3. 1. It would be very desirable to give a direct proof of the fullness of the G-spectrum of G−Locf A avoiding reference to the orbifold theory AG via the equivalence G−Locf A Locf AG S. This would amount to showing directly that g-localized transportable endomorphisms of A∞ exist for every g ∈ G. Since our proof relies on the fairly non-trivial modularity result for Locf AG , cf. [29] together with [64], this might turn out difficult. 2. In the VOA setting, Dong and Yamskulna [14] have shown that there exist twisted representations for all g ∈ G. Since [48, Prop. 3.23] is a purely categorical result, the above conclusion also holds in the VOA setting as soon as one can establish that the G-twisted representations form a rigid tensor category. 3. It may be useful to summarize the situation in a diagram:
Locf A 6 C;
CG
⊂ G−Locf A 6 C ;CS
? ? Locf AG ∩ S ⊂ Locf AG
The horizontal inclusions are full, Locf A being the degree zero subcategory of G − Locf A. If G is abelian, the G-grading passes to Locf AG (see [48]) and Locf AG ∩S is its degree zero subcategory. Moving from left to right or from top to bottom, the dimension of the categories are multiplied by |G|. In the upper line this is due to Theorem 4.2 and in the lower due to the results of [47]. Together with dim C = |G|·dim C S this implies dim Locf AG = |G|2 dim Locf A, as required by [29]. (In fact, this latter identity together with [48, Prop. 3.23] provides an alternative proof of the completeness of the G-spectrum of G−Locf A.) Furthermore, the upper left and lower right categories are modular, whereas Locf AG ∩S is not (whenever G = {e}). The passage Locf AG ∩S ; Locf A is the ‘modular closure’ from [44, 5] and Locf AG ∩ S ; Locf AG is the ‘minimal modularization’, conjectured to exist for every premodular category, cf. [47]. We briefly discuss the modularity of G−Locf A. In [60], a braided crossed G-category C was called modular if its braided degree zero subcategory Ce is modular in the usual sense [59]. This definition seems somewhat unsatisfactory since it does not take the nontrivially graded part of C into account. In [31], the vector space
VC =
i∈I g∈G
Hom(βg (Xi ), Xi ),
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where I indexes the isomorphism classes of simple objects in C, is introduced and an endomorphism S ∈ End VC is defined by its matrix elements u
S((X, u), (Y, v))) =
A A v
A X Y A
where ∂X = g, ∂Y = h and u : βh (X) → X, v : βg (Y ) → Y . A braided G-crossed fusion category is modular (in the sense of [31]) if the endomorphism S is invertible. Proposition 4.4. Let (H0 , A, ) be a completely rational chiral CFT and G a finite symmetry group. Then the braided crossed G-category G − Locf A is modular in the sense of [31]. Proof. As used above, the braided categories Locf A = (G−Locf A)e and Locf AG ∼ = (G−Locf A)G are modular. Now the claim follows by [31, Theorem 10.5]. The preceding discussions have been of a very general character. In the next subsection they will be used to elucidate completely the case of holomorphic orbifolds, where our results go considerably beyond (and partially diverge from) those of [11]. In the non-holomorphic case it is clear that comparably complete results cannot be hoped for. Nevertheless already a preliminary analysis leads to some surprising results and counterexamples, cf. the final subsection. 4.2. Orbifolds of holomorphic models. Definition 4.5. A holomorphic chiral CFT is a completely rational chiral CFT with trivial representation category Locf A. (I.e., Locf A is equivalent to Vect f C.) Remark 4.6. By the results of [29], a completely rational chiral CFT is holomorphic iff µ(A) = 1 iff A(E ) = A(E) whenever E = ∪ni=1 Ii , where Ii ∈ I with mutually disjoint closures. Corollary 4.7. Let A be a holomorphic chiral CFT acted upon by a finite group G. Then G−Locf A has precisely one isomorphism class of simple objects for every g ∈ G, all of these objects having dimension one. Proof. By Theorem 4.2, we have dim(G − Locf A)g = 1 for all g ∈ G. Since the dimensions of all objects are ≥ 1, the result is obvious.
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Remark 4.8. 1. In [43], where the invertible objects of G−Locf A were called soliton automorphisms, it is shown that these objects can be studied in a purely local manner. 2. Let A be a holomorphic chiral CFT, and pick an interval I ∈ K. By Corollary 4.7 there is just one isoclass of simple objects in (G−Locf A)g for every g ∈ G. Since the objects of G−Locf A are transportable endomorphisms of A∞ , we can pick, for every g ∈ G, representer ρg that is g-localized in I . By Lemma 2.12, ρg restricts to an automorphism of A(I ). Furthermore, we can choose unitaries ug,h ∈ HomA(I ) (ρg ρh , ρgh ). In other words, we have a homomorphism G → AutA(I )/InnA(I ) =: OutA(I ),
g → [ρg ],
thus a ‘G-kernel’, cf. [57]. We recall some well known facts: The associativity (ρg ρh )ρk = ρg (ρh ρk ) implies the existence of αg,h,k ∈ T such that ugh,k ug,h = αg,h,k ug,hk ρg (uh,k )
∀g, h, k.
A tedious but straightforward computation using four ρ’s shows that α : G×G×G → T is a 3-cocycle, whose cohomology class [α] ∈ H 3 (G, T) does not depend on the choice of the ρ’s and of the u’s. Thus [α] is an obstruction to the existence of representers ρg for which g → ρg is a homomorphism G → AutA(I ). (Actually, since in QFT the algebras A(I ) are type III factors with separable predual, the converse is also true: If [α] = 0 then one can find a homomorphism g → ρg , cf. [57].) 4.9. For a further analysis it is more convenient to adopt a purely categorical viewpoint. Starting with the category G − Locf A of a holomorphic theory A, we don’t lose any information by throwing away the non-simple objects and the zero morphisms. In this way we obtain a categorical group C, i.e. a monoidal groupoid where all objects have a monoidal inverse. The set of isoclasses is the group G. In the general k-linear case it is well known that such categories are classified up to equivalence by H 3 (G, k ∗ ). This i.e. a full subcategory is shown by picking an equivalent skeletal tensor category C, with one object per isomorphism class. Even if C is strict, C in general is not, and the associativity constraint defines an element of H 3 (G, k ∗ ). It is thus clear that 3-cocycles on G will play a rˆole in the classification of the braided crossed G-categories associated with holomorphic QFTs. In view of [11, 10, 12] and [13, 14] this is hardly surprising. Yet, the situation is somewhat more involved than anticipated by most authors since a classification of the possible categories G − Locf A – and therefore of the categories Locf AG – must also take the G-action on G−Locf A and the braiding into account. If one considers braided categorical groups, G must be abelian and one has a classifi3 (G, k ∗ ), cf. [25]. (H 3 (G, k ∗ ) is Mac Lane’s cohomology [38] for cation in terms of Hab ab abelian groups.) The requirement that G be abelian disappears if one admits a non-trivial G-action and considers braided crossed G-categories. One finds [60] that (non-strict) skeletal braided crossed G-categories with strict G-action in the sense of (3.3) are clas3 (G, k ∗ ) [51]. Unfortunately, this sified in terms of Ospel’s quasiabelian cohomology Hqa is still not sufficient for our purposes. Namely, assume we have a braided crossed G-category C that is also a categorical group (and thus a categorical G-crossed module in the sense of [7]). Even if C is strict monoidal and satisfies (3.3) – as our categories G−Loc A and C S do – an equivalent skeletal category C in general will not satisfy (3.3). It is clear that for a completely general classification of braided crossed G-categories that are categorical groups one can proceed along similar lines as in the classifications cited above. We will supply the details in the near future [49], also elucidating the rˆole of the twisted quantum doubles D ω (G) [10] in the present context. (Note that the modular
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category D ω − Mod contains the symmetric category G − Mod as a full subcategory, and D ω − Mod G − Mod is a braided crossed G-category with precisely one invertible object of every degree. However, not every such category is equivalent to D ω − Mod G − Mod for some [ω] ∈ H 3 (G, T)!)
4.3. Some observations on non-holomorphic orbifolds. In the previous subsection we have seen that a holomorphic chiral CFT A has (up to isomorphism) exactly one simple object of degree g ∈ G, and this object has dimension one, thus is invertible. This allows a complete classification of the categories G−LocA and LocAG ∼ = (G−LocA)G that can arise. It is clear that in the non-holomorphic case (Locf A Vect C ) there is no hope of obtaining results of this completeness. The best one could hope for would be a classification of the categories G−Locf A that can arise from CFTs with prescribed Locf A (G − Locf A)e , but for the time being this is far out of reach. We therefore content ourselves with some comments on a more modest question. To wit, we ask whether a non-holomorphic completely rational CFT A admits invertible g-twisted representations for every g ∈ G. (As we have seen, this is the case for holomorphic A.) It turns out that the existence of a braiding (in the sense of crossed G-categories) provides an obstruction: Lemma 4.10. Let C be a braided crossed G-category. If there exists an invertible object of degree g ∈ G then γg (X) ∼ = X ∀X ∈ Ce . Proof. Let X ∈ Ce and Y ∈ Cg . Then the braiding gives rise to isomorphisms cX,Y : X ⊗ Y → Y ⊗ X and cY,X : Y ⊗ X → γg (X) ⊗ Y . Composing these we obtain an isomorphism X ⊗ Y → γg (X) ⊗ Y . If Y is invertible, we can cancel it by tensoring with ∼ =
Y , obtaining the desired isomorphism X −→ γg (X).
Corollary 4.11. Let C be a braided crossed G-category and let g ∈ G. If there exists X ∈ Ce such that γg (X) ∼ X then there exists no invertible Y ∈ Cg . = Remark 4.12. The condition γg (X) ∼ = X ∀X ∈ Ce is necessary in order for the existence of invertible objects of degree g, but of course not sufficient. In any case, there are many chiral CFTs where the corollary, as applied to G−Locf A, excludes invertible g-twisted representations for g = e. One such class will be considered below. We apply the above results to the n-fold direct product A = B ⊗n of a completely rational chiral CFT B, on which the symmetric group Sn acts in the obvious fashion. We first note that every irreducible π ∈ Repf A is unitarily equivalent to a direct product π1 ⊗ · · · ⊗ πn of irreducible πi ∈ Repf B, cf. [29]. Thus the equivalence classes of simple objects of Locf A are the n-tuples of equivalence classes of simple objects of Locf B, and Sn acts on them by permutation. Corollary 4.13. Let B be a completely rational chiral CFT and let n ≥ 2. Consider A = B ⊗n with the permutation action of G = Sn . If B is not holomorphic then G−Locf A contains no invertible object ρ with ∂ρ = e.
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Proof. Since B is not holomorphic we can find a simple object σ ∈ Locf B such that σ ∼ 1. If g ∈ Sn with g = e there is i ∈ {1, . . . , n} such that g(i) = i. Consider an = object ρ = (ρ1 , . . . , ρn ) ∈ Locf A, where ρi = 1 and ρg(i) = σ . Now it is clear that γg (ρ) ∼ ρ, and Corollary 4.11 applies. = For any tensor category C we denote by Pic(C) the full monoidal subcategory of invertible objects. (In a ∗-category these are precisely the objects of dimension one.) Corollary 4.14. Let B be a completely rational chiral CFT. Consider A = B ⊗n for n ≥ 2 and let G ⊂ Sn be a subgroup. If B is non-holomorphic then Pic(Locf AG ) ∼ = Pic (Locf A)G . Proof. We may assume G = {e} since otherwise there is nothing to prove. By Theorem 3.12 we have Locf AG ∼ = (G − Locf A)G . Let now ρ ∈ Pic(Locf AG ). Then G E(ρ) ∈ Pic((G−Locf A) ), and by Corollary 4.13 we have ∂E(ρ) = e, thus E(ρ) ∈ Pic((Locf A)G ). The rest follows as in Subsect. 3.4. Thus, in permutation orbifold models, the Picard category Pic(Locf AG ) is determined already by Pic(Locf A) and the G-action on it, i.e. we do not need to know the g-twisted representations of A for g = e. We recall that a subgroup G ⊂ Sn is called transitive if for each i, j ∈ {1, . . . , n} there exists g ∈ G such that g(i) = j . Corollary 4.15. Let B be a non-holomorphic completely rational chiral CFT. Consider A = B ⊗n for n ≥ 2 and let G ⊂ Sn be a transitive subgroup. Then the isomorphism classes in Pic(Locf AG ) are in 1-1 correspondence with the pairs ([σ ], λ), where [σ ] is 1 = G an isomorphism class in Pic(Locf B) and λ ∈ G ab is a one-dimensional character of G. Proof. Let ρ be an invertible object of Locf AG . By Corollary 4.14, we have E(ρ) ∼ = (σ1 , . . . , σn ) where the σi are invertible objects of Locf B. By Subsect. 3.2, E(ρ) is invariant under the G-action on Locf A, and since the latter transitively permutes the σi there is σ ∈ Pic(Locf B) such that σi ∼ = σ for all i. Now, by 3.3 we know that for 1 there exist localized unitaries uλ ∈ A∞ such that βg (uλ ) = λ(g)uλ . In every λ ∈ G restriction to AG ∞ , the localized isomorphisms Ad uλ are inequivalent invertible objects ρλ ∈ Pic(Locf AG ). Now the claimed bijection follows by picking one representer σ for each isoclass [σ ] in Pic(Locf B) and mapping ([σ ], λ) → [(σ, . . . , σ ) ⊗ ρλ ]. Remark 4.16. Corollary 4.15 is incompatible with some claims in [1] on the representation category of cyclic orbifolds, for which no rigorous proofs are available, cf. also [39]. More precisely, in [1, 2] cyclic permutation orbifolds (B ⊗n )Zn are considered, where Zn ⊂ Sn acts by cyclic permutations on A = B ⊗n , thus transitively. According to [1, 2] the set of isoclasses of simple objects of Repf AZn contains a subset I = {([X], i, j )}, where [X] is a simple isoclass in Repf B and i, j ∈ Z/nZ. Accepting this for a minute, it follows from the modular S-matrix given in [2] that if X is invertible (‘X is a simple current of B’), also the objects ([X], i, j ) in Repf AZn are invertible. Thus according to [1, 2], Repf AZn contains at least n2 m isoclasses of invertible objects, where m is the number of isoclasses of invertible objects of Repf B. On the other hand, Corollary 4.15 implies that Repf AZn contains precisely nm isoclasses of invertible objects. Since n ≥ 2 and m ≥ 1 this is a contradiction.
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The claims in question play a crucial rˆole in [2] which attempts to prove that the kernel of the representation of SL(2, Z) arising from a rational CFT contains a congruence subgroup (N ). We leave it to the reader to draw his/her conclusions. Acknowledgements. The research reported here was presented at the workshop ‘Tensor Categories in Mathematics and Physics’ which took place at the Erwin Schr¨odinger Institute, Vienna, in June 2004. I am grateful to the ESI for hospitality and financial support and to the organizers for the invitation to a very stimulating meeting.
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Commun. Math. Phys. 260, 763–763 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1422-6
Communications in
Mathematical Physics
Erratum Conformal Orbifold Theories and Braided Crossed G-Categories Michael Muger ¨ Department of Mathematics, Radboud University, Nijmegen, The Netherlands. E-mail: [email protected] Received: 14 March 2005 / Accepted: 14 March 2005 Erratum published online: 11 October 2005 – © Springer-Verlag 2005 Commun. Math. Phys. 260, 727–762 (2005)
In the final Remark 4.16 I claimed that the results of its Subsect. 4.3 are in contradiction to what can be derived from certain statements in [2], which in turn follow from [1]. This claim was wrong, being based on an erroneous deduction from the statements in [1, 2]. I regret this mistake. In fact, Bantay has provided me with a convincing argument to the effect that my Corollary 4.15 can also be deduced from the results of his earlier article [1]. His argument relies on the formula [1, Eq. (15)] for the S-matrix of the permutation orbifold, which in turn follows from the character formula [1, Eq. (5)]. In view of the above, the last sentence of the abstract and the penultimate sentence of the introduction should read: “We conclude with some partial results on non-holomorphic orbifolds.” References 1. Bantay, P.: Characters and modular properties of permutation orbifolds. Phys. Lett. B419, 175–178 (1998). Permutation orbifolds. Nucl. Phys. B633, 365–378 (2002) 2. Bantay, P.: The kernel of the modular representation and the Galois action in RCFT. Commun. Math. Phys. 233, 423–438 (2003) Communicated by Y. Kawahigashi
Supported by NWO.
